Permutations and Combinations
144 JEE Maths previous year questions on Permutations and Combinations — options free on every question; 14 include the answer & explanation free, the rest unlock with PYQ Pass.
The largest \(\mathrm{n}\in \mathrm{N}\) such that \({3}^{\mathrm{n}}\) divides \(50\) ! is:
[JEE Main 2025, 2 Apr (Shift 1)]
\(22\)
To find the exponent of a prime \(p\) in \(n!\), we use:
\(\text{Exponent of }p\text{ in }n!=\sum _{k=1}^{\mathrm{∞}}⌊\frac{n}{{p}^{k}}⌋\)
Here, \(p=3\), and \(n=50\). So,
\(\sum _{k=1}^{\mathrm{∞}}⌊\frac{50}{{3}^{k}}⌋=⌊\frac{50}{3}⌋+⌊\frac{50}{9}⌋+⌊\frac{50}{27}⌋+⌊\frac{50}{81}⌋+\ldots\)
Now compute:
- \(⌊\frac{50}{3}⌋=16\)
- \(⌊\frac{50}{9}⌋=5\)
- \(⌊\frac{50}{27}⌋=1\)
- \(⌊\frac{50}{81}⌋=0\)
Adding up:
\(16+5+1=22\)
Answer: \(22\)Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:
[JEE Main 2025, 24 Jan (Shift 2)]
8925
Group -A Group -B
4B + 1G OB + 3G
3B + 2G 1B + 2G
2B + 3G 2B + 1G
\(Totalpossiblecases\\ ={}^{7}\mathrm{C}_{4}\times {}^{3}\mathrm{C}_{1}\times {}^{6}\mathrm{C}_{0}\times {}^{5}\mathrm{C}_{3}+{}^{7}\mathrm{C}_{3}\times {}^{3}\mathrm{C}_{2}\times {}^{6}\mathrm{C}_{1}\\ \times {}^{5}\mathrm{C}_{2}+{}^{7}\mathrm{C}_{2}\times {}^{3}\mathrm{C}_{3}\times {}^{6}\mathrm{C}_{2}\times {}^{5}\mathrm{C}_{1}\\ =\frac{7\times 6\times 5}{3\times 2}\times 3\times 1\times \frac{5\times 4}{2}+\frac{7\times 6\times 5}{3\times 2}\\ \times 3\times 6\times \frac{5\times 4}{2}+\frac{7\times 6}{2}\times 1\times \frac{6\times 5}{2}\times 5\\ =1050+6300+1575\\ =8925\)
Let \({P}_{n}\) denote the total number of triangles formed by joining the vertices of an \(n -\)side regular polygon. If \({P}_{n+1}-{P}_{n}=66\), then the sum of all distinct prime divisors of \(n\) is:
[JEE Main 2026, 2 Apr (Shift 2)]
\(5\)
\( P_n=\) number of triangles formed by joining the vertices of an \(n -\)side regular polygon.
\( P_n={ }^n C_3 \)
\( { }^{n+1} C_3-{ }^n C_3=66\)
\( \Rightarrow \frac{(\mathrm{n}+1)(\mathrm{n})(\mathrm{n}-1)}{6}-\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{6}=66\)
\( \Rightarrow \mathrm{n}(\mathrm{n}-1)[(\mathrm{n}+1)-(\mathrm{n}-2)]=396 \)
\( \Rightarrow 3 \mathrm{n}(\mathrm{n}-1)=396\)
\( \mathrm{n}^2-\mathrm{n}-132=0 \)
\( \Rightarrow(\mathrm{n}-12)(\mathrm{n}+11)=0\)
\( \Rightarrow \mathrm{n}=12,-11 \)
\( 12=2^2 3^1\)
divisor are \(=1,2,3,4,6,12\)
prime divisor are \(=3,2 \)
sum \(=5\)
The largest value of \(n\), for which \(40^n\) divides \(60!\), is
[JEE Main 2026, 24 Jan (Shift 2)]
14
\({40}^{n}={2}^{3n}\times {5}^{n}\)
Now, highest power of 2 which divides \(60!\) is
\({E}_{2}\left(60!\right)=\left[\frac{60}{2}\right]+\left[\frac{60}{{2}^{2}}\right]+\left[\frac{60}{{2}^{3}}\right]+\left[\frac{60}{{2}^{4}}\right]+\left[\frac{60}{{2}^{5}}\right]\)
\(=30+15+7+3+1=56\)
Now, highest power of 5 which divides \(60!\) is
\({E}_{5}\left(60!\right)=\left[\frac{60}{5}\right]+\left[\frac{60}{{5}^{2}}\right]\)
\(=12+2=14\)
\({40}^{n}={\left({2}^{3}\right)}^{n}\times {5}^{n}={\left({2}^{3}\times 5\right)}^{n}\)
\(60!={2}^{56}\times {5}^{14}\ldots ={2}^{14}⋅{\left({2}^{3}⋅5\right)}^{14}\)
Maximum value of \(n\) is \(14\).
The letters of the word "\(UDAYPUR\)" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "\(UDAYPUR\)" is
[JEE Main 2026, 24 Jan (Shift 2)]
\(1580\)
Dictionary order is \(A,D, P, R, U, Y\)
number of words starting with \(A\to \frac{6!}{2!}=360\)
number of words starting with \(D\to \frac{6!}{2!}=360\)
number of words starting with \(P\to \frac{6!}{2!}=360\)
number of words starting with \(R\to \frac{6!}{2!}=360\)
number of words starting with \(U A \rightarrow 5!=120\)
number of words starting with \(UDAP\to 3!=6\)
number of words starting with \(UDAR\to 3!=6\)
number of words starting with \(UDAU\to 3!=6\)
number of words starting with \(UDAYPRU\to 1\)
number of words starting with \(UDAYPUR\to 1\)
Total \(=1580\)
How many words can be formed from the word DAUGHTER such that any vowels are not together
\(36000\)
Required number of ways=
$$\begin{aligned}& 8!-6!\times 3! \\& =36,000\end{aligned}$$
In a group of 3 girls and 4 boys, there are two boys \({B}_{1}and{B}_{2}\). The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \({B}_{1}and{B}_{2}\) are not adjacent to each other, is :
[JEE Main 2025, 22 Jan (Shift 2)]
144
\(\text{Required number of ways}=\\ \text{Total ways}-\text{ways in which}{\mathrm{B}}_{1}\text{and}{\mathrm{B}}_{2}\text{are together}\\ =2!(3!4!)-2!(3!(3!2!))=144\)
There are 5 points \({P}_{1},{P}_{2},{P}_{3},{P}_{4},{P}_{5}\) on the side \(AB\), excluding \(A\) and \(B\), of a triangle \(ABC\). Similarly there are 6 points \({P}_{6},{P}_{7},\ldots ,{P}_{11}\) on the side BC and 7 points \({P}_{12},{P}_{13},\ldots ,{P}_{18}\) on the side CA of the triangle. The number of triangles, that can be formed using the points \({P}_{1},{P}_{2},\ldots ,{P}_{18}\) as vertices, is :
[JEE Main 2024, 4 Apr (Shift 1)]
751
Total ways of selecting \(3\) points out of \(18\) is \({ }^{18} \mathrm{C}_3\).
Total ways of selecting \(3\) points from \(\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \mathrm{P}_4, \mathrm{P}_5\) is \({ }^5 \mathrm{C}_3\).
Total ways of selecting \(3\) points from \(\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}\) is \({ }^6 \mathrm{C}_3\).
Total ways of selecting \(3\) points from \(\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}\) is \({ }^7 \mathrm{C}_3\).
Hence,The number of triangles, that can be formed using the points \(\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}\) is
\({ }^{18} C_3-{ }^5 C_3-{ }^6 C_3-{ }^7 C_3 \)
\(=751\)
A box contains \(5\) blue, \(6\) yellow and \(4\) red balls. The number of ways of drawing \(8\) balls containing at least two balls of each colour is:
[JEE Main 2026, 5 Apr (Shift 2)]
\(4100\)
The possible selections of balls are:
\((2\) blue, \(2\) yellow, \(4\) red), ( \(2\) blue, \(4\) yellow, \(2\) red),
( \(4\) blue, \(2\) yellow, \(2\) red), ( \(3\) blue, \(3\) yellow, \(2\) red),
( \(3\) blue, \(2\) yellow, \(3\) red) or ( \(2\) blue, \(3\) yellow, \(3\) red)
\(={ }^5 C_2 \cdot{ }^6 C_2 \cdot{ }^4 C_4+{ }^5 C_2 \cdot{ }^6 C_4 \cdot{ }^4 C_2+{ }^5 C_4 \cdot{ }^6 C_2 \cdot{ }^4 C_2+{ }^5 C_3 \cdot{ }^6 C_3\). \({ }^4 C_2+{ }^5 C_3 \cdot{ }^6 C_2 \cdot{ }^4 C_3+{ }^5 C_2 \cdot{ }^6 C_3 \cdot{ }^4 C_3\)
\(=150+900+450+1200+600+800 =4100\)
The number of 3 digit numbers which is divisible
by 2 and 3 but not divisible by 4 and 9.
(24 Jan, Shift I, Memory Based)
125
If number is divisible by 4 & 9 then is divisible by 36.
Total 3 digit numbers are 900
Required 3 digit number = Total 3 digit number which are divisible by 6 - total 3 digit number which are divisible by 36
= \(\frac{900}{6} - \frac{900}{36}\) = 125
The letters of the word "\(UDAYPUR\)" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "\(UDAYPUR\)" is
[JEE Main 2026, 24 Jan (Shift 2)]
\(1580\)
Dictionary order is \(A,D, P, R, U, Y\)
number of words starting with \(A\to \frac{6!}{2!}=360\)
number of words starting with \(D\to \frac{6!}{2!}=360\)
number of words starting with \(P\to \frac{6!}{2!}=360\)
number of words starting with \(R\to \frac{6!}{2!}=360\)
number of words starting with \(U A \rightarrow 5!=120\)
number of words starting with \(UDAP\to 3!=6\)
number of words starting with \(UDAR\to 3!=6\)
number of words starting with \(UDAU\to 3!=6\)
number of words starting with \(UDAYPRU\to 1\)
number of words starting with \(UDAYPUR\to 1\)
Total \(=1580\)
Line \({L}_{1}\) of slope \(2\) and line \({L}_{2}\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \({\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots .{\mathrm{P}}_{12}\) are 12 points on line \({L}_{1}\) and \({Q}_{1},{Q}_{2},\ldots ..{Q}_{9}\) are \(9\) points on line \({\mathrm{L}}_{2}\). Then the total number of triangles, that can be formed having vertices at three of the \(22\) points \(\mathrm{O},{\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots {\mathrm{P}}_{12}\), \({\mathrm{Q}}_{1},{\mathrm{Q}}_{2},\ldots .{\mathrm{Q}}_{9}\), is:
\(1134\)
Total number of \(\Delta\) are
\(={ }^9 \mathrm{C}_1{ }^{12} \mathrm{C}_2+{ }^9 \mathrm{C}_2{ }^{12} \mathrm{C}_1+{ }^1 \mathrm{C}_1{ }^9 \mathrm{C}_1{ }^{12} \mathrm{C}_1 \)
\(=594+432+108 \)
\(=1134\)
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
[JEE Main 2024, 06 Apr (Shift 1)]
16
\(∵\) No. of triangles having no side common with an sided polygon
\(=\frac{{}^{n}{C}_{1}{.}^{n−4}{C}_{2}}{3}\)
\(=\frac{{}^{8}{C}_{1}{.}^{4}{C}_{2}}{3}=16\)
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
[JEE Main 2024, 06 Apr (Shift 1)]
16
\(∵\) No. of triangles having no side common with an sided polygon
\(=\frac{{}^{n}{C}_{1}{.}^{n−4}{C}_{2}}{3}\)
\(=\frac{{}^{8}{C}_{1}{.}^{4}{C}_{2}}{3}=16\)
How many 6 letter words can be formed using the word "MATHS'' such that any letter can be used maximum two times?
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Let \(S=\{1,2,3,4,5,6,7,8,9\}\). Let \(x\) be the number of 9-digit numbers formed using the digits of the set \(S\) such that only one digit is repeated and it is repeated exactly twice. Let \(y\) be the number of 9-digit numbers formed using the digits of the set \(S\) such that only two digits are repeated and each of these is repeated exactly twice. Then,
[JEE Main 2026, 28 Jan (Shift 1)]
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If \(S={2,4,8,16,...,512}\) If S is broken in 3 equal subsets A, B and C such that \(A\cap B=B\cap C=C\cap A=ϕ\) and \(A\cup B\cup C=S\), then maximum number of ways to break is:
[JEE Main 2024]
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Line \({L}_{1}\) of slope \(2\) and line \({L}_{2}\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \({\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots .{\mathrm{P}}_{12}\) are 12 points on line \({L}_{1}\) and \({Q}_{1},{Q}_{2},\ldots ..{Q}_{9}\) are \(9\) points on line \({\mathrm{L}}_{2}\). Then the total number of triangles, that can be formed having vertices at three of the \(22\) points \(\mathrm{O},{\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots {\mathrm{P}}_{12}\), \({\mathrm{Q}}_{1},{\mathrm{Q}}_{2},\ldots .{\mathrm{Q}}_{9}\), is:
[JEE Main 2025, 3 Apr (Shift 2)]
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If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at \(440^{\text {th }}\) position in this arrangement, is :
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A triangle PQR is such that 3 points lie on the side PQ,4 points on QR and 5 points on RP respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?
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Let the set \(S={2,4,8,16,\ldots ,512}\) be partitioned into 3 sets \(A, B, C\) with equal number of elements such that \(A\cup B\cup C=S\) and \(A\cap B=B\cap C=A\cap C=ϕ\). The maximum number of such possible partitions of \(S\) is equal to :
[JEE Main 2024, 5 Apr (Shift 2)]
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Numbers of 7 letter words formed by the numbers "1,2,3" such that the sum of digits is 11.
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There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is
[JEE Main 2025, 8 Apr (Shift 1)]
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Let \(\mathrm{A}={(\mathrm{a},\mathrm{b},\mathrm{c}):\mathrm{a},\mathrm{b},\mathrm{c}\) are non-negative integers and \(\mathrm{a}+\mathrm{b}+2\mathrm{c}=22}\). Then \(n(\mathrm{A})\) is equal to:
[JEE Main 2026, 4 Apr (Shift 2)]
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The number of ways, of forming a queue of \(4\) boys and \(3\) girls such that all the girls are not together, is:
[JEE Main 2026, 4 Apr (Shift 1)]
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The largest \(\mathrm{n}\in \mathrm{N}\) such that \({3}^{\mathrm{n}}\) divides \(50!\) is:
[JEE Main 2025, 2 Apr (Shift 1)]
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The number of 3 digit numbers which is divisible
by 2 and 3 but not divisible by 4 and 9.
(24 Jan, Shift I, Memory Based)
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The number of ways in which 5 girls and 3 boys can be seated in a row so that no two boys are together is
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Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:
[JEE Main 2025, 24 Jan (Shift 2)]
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If the sum \(\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \cdot 2^{29}\), then \(\alpha=\) (22 Jan, Shift II, Memory Based)
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In a group of 3 girls and 4 boys, there are two boys \(B_1\) and \(B_2\). The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \(B_1\) and \(B_2\) are not adjacent to each other, is :
[JEE Main 2025, 22 Jan (Shift 2)]
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The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :
[JEE Main 2024, 8 Apr (Shift 2)]
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\(\text{ If the letters of the word " KANPUR" are arranged in}\\ \text{dictionary, then the }{440}^{\text{th }}\text{ word is }\)
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Let \(P\) be the set of seven digit numbers with sum of their digits equal to \(11.\) If the numbers in \(P\) are formed by using the digits \(1,2\) and \(3\) only, then the number of elements in the set \(P\) is
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Let \(p\)be the number of all triangles that can be formed by joining the vertices of a regular polygon\(P\)of \(n\)-sides and \(q\)be the number of all quadrilaterals that can be formed by joining the vertices of \(P\). If \(p+q=126\), then the eccentricity of the ellipse \(\frac{{x}^{2}}{16}+\frac{{y}^{2}}{n}=1\) is :
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Number of ways to form 5 digit numbers greater than 50000 with the use of digits 0,1,2,3,4,5, 6,7 such that sum of first and last digit is not more than 8, is equal to (28 Jan, Shift I, Memory Based)
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The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is
[JEE Main 2025, 28 Jan (Shift 1)]
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4 boys and 3 girls are to be seated in a row such that all girls seat together and all boys seat together and two particular boys \(B_1\) and \(B_2\) are not adjacent to each other. Then the number of ways in which this arrangement can be done. (22 Jan, Shift II, Memory Based)
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Let \(P\) be the set of seven digit numbers with sum of their digits equal to \(11.\) If the numbers in \(P\) are formed by using the digits \(1,2\) and \(3\) only, then the number of elements in the set \(P\) is
[JEE Main 2025, 29 Jan (Shift 1)]
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A student has to answer 10 questions, choosing at least 4 from each of the parts \(A\) and \(B\). If there are 6 questions in part \(A\) and 7 in part \(B\), then the number of ways can the student choose 10 questions is
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The largest value of \(n\), for which \(40^n\) divides \(60!\), is
[JEE Main 2026, 24 Jan (Shift 2)]
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If \(n\) is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then \(n\) is equal to:
[JEE Main 2024, 1 Feb (Shift 1)]
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Numbers of 7 letter words formed by the numbers "1,2,3" such that the sum of digits is 11.
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Number of ways to form 5 digit numbers greater than 50000 with the use of digits 0,1,2,3,4,5, 6,7 such that sum of first and last digit is not more than 8, is equal to (28 Jan, Shift I, Memory Based)
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\({ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}\) if and only if :
[JEE Main 2024, 27 Jan (Shift 1)]
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The number of arrangements of all digits of 12345 such that at least 3 digits will not come in its position is:
[JEE Main 2024]
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If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at \(440^{\text {th }}\) position in this arrangement, is :
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The number of ways, in which \(16\) oranges can be distributed to four children such that each child gets at least one orange, is
[JEE Main 2026, 23 Jan (Shift 2)]
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If the sum \(\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \cdot 2^{29}\), then \(\alpha=\) (22 Jan, Shift II, Memory Based)
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At an election, a voter may vote for any number of candidates not exceeding the number to be elected. If \(\mathbf{4}\) candidates are to be elected out of the \(\mathbf{1 2}\) contested in the election and voter votes for at least one candidate, then the number of ways of selections is:
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The number of ways, in which \(16\) oranges can be distributed to four children such that each child gets at least one orange, is
[JEE Main 2026, 23 Jan (Shift 2)]
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How many words can be formed from the word DAUGHTER such that any vowels are not together
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There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is
[JEE Main 2025, 8 Apr (Shift 1)]
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A 5-letter word is to be made using any distinct 5 alphabets such that middle alphabet is \(M\) and letters should be in increasing order. (22 Jan, Shift I, Memory Based)
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A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:
[JEE Main 2026, 8 Apr (Shift 2)]
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The number of words, which can be formed using all the letters of the word “DAUGHTER”, so that all the vowels never come together, is
[JEE Main 2025, 23 Jan (Shift 1)]
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If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at \(440^{\text {th }}\) position in this arrangement, is :
[JEE Main 2025, 29 Jan (Shift 2)]
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At an election, a voter may vote for any number of candidates not exceeding the number to be elected. If \(\mathbf{4}\) candidates are to be elected out of the \(\mathbf{1 2}\) contested in the election and voter votes for at least one candidate, then the number of ways of selections is:
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Let \({}^{n}C_{r-1}=28,{}^{n}C_{r}=56\) and \({}^{n}C_{r+1}=70\), let \(A(4cost,4\sin t)\), \(B(2\sin t,-2cost)\) and \(C\left(3r-n,{r}^{2}-n-1\right)\) be the vertices of a triangles ABC, where \(t\) is a parameter. If \((3x-1{)}^{2}+(3y{)}^{2}=\alpha\), is the locus of the centroid of triangle ABC , then \(\alpha\) equates. (28 Jan, Shift I, Memory Based)
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Let \(p\)be the number of all triangles that can be formed by joining the vertices of a regular polygon \(P\)of \(n\)-sides and \(q\)be the number of all quadrilaterals that can be formed by joining the vertices of \(P\). If \(p+q=126\), then the eccentricity of the ellipse \(\frac{{x}^{2}}{16}+\frac{{y}^{2}}{n}=1\) is :
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Let \(\alpha=\frac{(4 !) !}{(4 !)^{3 !}}\) and \(\beta=\frac{(5 !) !}{(5 !)^{4 !}}\). Then :
[JEE Main 2024, 27 Jan (Shift 2)]
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Let \(\alpha=\frac{(4 !) !}{(4 !)^{3 !}}\) and \(\beta=\frac{(5 !) !}{(5 !)^{4 !}}\). Then :
[JEE Main 2024, 27 Jan (Shift 2)]
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How many 6 letter words can be formed using the word "MATHS'' such that any letter can be used maximum two times?
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The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is
[JEE Main 2025, 28 Jan (Shift 1)]
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Let \(S\) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \(S\), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :
[JEE Main 2025, 28 Jan (Shift 2)]
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If \(S={2,4,8,16,...,512}\) If S is broken in 3 equal subsets A, B and C such that \(A\cap B=B\cap C=C\cap A=ϕ\) and \(A\cup B\cup C=S\), then maximum number of ways to break is:
[JEE Main 2024]
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If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at \(440^{\text {th }}\) position in this arrangement, is :
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Let \({}^{n}C_{r-1}=28,{}^{n}C_{r}=56\) and \({}^{n}C_{r+1}=70\), let \(A(4cost,4\sin t)\), \(B(2\sin t,-2cost)\) and \(C\left(3r-n,{r}^{2}-n-1\right)\) be the vertices of a triangles ABC, where \(t\) is a parameter. If \((3x-1{)}^{2}+(3y{)}^{2}=\alpha\), is the locus of the centroid of triangle ABC , then \(\alpha\) equates. (28 Jan, Shift I, Memory Based)
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Number of ways of arranging \(8\) identical books into \(4\) identical shelves where any number of shelves may remain empty is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
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Number of ways of arranging \(8\) identical books into \(4\) identical shelves where any number of shelves may remain empty is equal to
[JEE Main 2024, 29 Jan (Shift 2)]
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A building has ground floor and \(10\) more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the \(10\)th floor. The number of ways, in which any \(4\) persons exit at a floor and the remaining \(5\) persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to:
[JEE Main 2026, 6 Apr (Shift 2)]
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The number of \(4-\)letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word \(\mathrm{INCONSEQUENTIAL}\), without repeating any letter, is:
[JEE Main 2026, 6 Apr (Shift 1)]
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4 boys and 3 girls are to be seated in a row such that all girls seat together and all boys seat together and two particular boys \(B_1\) and \(B_2\) are not adjacent to each other. Then the number of ways in which this arrangement can be done. (22 Jan, Shift II, Memory Based)
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The number of words, which can be formed using all the letters of the word “DAUGHTER”, so that all the vowels never come together, is
[JEE Main 2025, 23 Jan (Shift 1)]
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A 5-letter word is to be made using any distinct 5 alphabets such that middle alphabet is \(M\) and letters should be in increasing order. (22 Jan, Shift I, Memory Based)
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\(\text{ If the letters of the word " KANPUR" are arranged in}\\ \text{dictionary, then the }{440}^{\text{th }}\text{ word is }\)
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The number of seven-digit numbers, that can be formed by using the digits \(1, 2, 3, 5\) and \(7\) such that each digit is used at least once, is:
[JEE Main 2026, 2 Apr (Shift 1)]
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Let \(S=\{1,2,3,4,5,6,7,8,9\}\). Let \(x\) be the number of 9 -digit numbers formed using the digits of the set \(S\) such that only one digit is repeated and it is repeated exactly twice. Let \(y\) be the number of 9-digit numbers formed using the digits of the set \(S\) such that only two digits are repeated and each of these is repeated exactly twice. Then,
[JEE Main 2026, 28 Jan (Shift 1)]
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Let \(S={{x}^{3}+a{x}^{2}+bx+c:a,b,c\in N\) and \(a,b,c\leq 20}\) be a set of polynomials. Then the number of polynomials in \(S\), which are divisible by \(x^2+2\), is
[JEE Main 2026, 28 Jan (Shift 1)]
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Let \(S={{x}^{3}+a{x}^{2}+bx+c:a,b,c\in N\) and \(a,b,c\leq 20}\) be a set of polynomials. Then the number of polynomials in \(S\), which are divisible by \(x^2+2\), is
[JEE Main 2026, 28 Jan (Shift 1)]
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The number of arrangements of all digits of 12345 such that at least 3 digits will not come in its position is:
[JEE Main 2024]
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The sum of all the 4 -digit distinct numbers that can be formed with the digits \(1,2,2\) and 3 is:
[JEE Main 2021, 18 Mar (Shift 1)]
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The number of integers, greater than \(7000\) that can be formed, using the digits \(3,5,6,7,8\) without repetition, is
[JEE Main 2023, 24 Jan (Shift 2)]
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If the digits are not allowed to repeat in any number formed by using the digits \(0,2,4,6,8\), then the number of all numbers greater than 10,000 is equal to......
[JEE Main 2021, 22 Jul (Shift 2)]
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The number of ways of selecting two numbers \(a\) and \(b, a\) \(\in\{2,4,6, \ldots ., 100\}\) and \(b \in\{1,3,5, \ldots ., 99\}\) such that 2 is the remainder when \(a+b\) is divided by 23 is
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The total number of positive integral solutions \( (x, y, z) \) such that \( x y z=24 \) is
[JEE Main 2021, 25 Feb (Shift 1)]
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The number of triplets \((x, y, z)\). where \(x, y, z\) are distinct non negative integers satisfying \(x+y+z=15\), is
[JEE Main 2023, 11 Apr (Shift 1)]
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If \(a, b, c\) are three natural numbers in AP and \(\mathrm{a}+b+c=21\) then the possible number of values of the ordered triplet \((a, b, c)\) is
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If the sides \( A B, B C \) and \( C A \) of a triangle \( A B C \) have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :
[JEE Main 2021, 17 Mar (Shift 2)]
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There are \(15\) players in a cricket team, out of which \(6\) are bowlers, \(7\) are batsman and \(2\) are wicketkeepers. The number of ways, a team of \(11\) players be selected from them so as to include at least \(4\) bowlers, \(5\) batsman and \(1\) wicketkeeper, is.
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Total number of ways of selecting 2 white squares on a normal chess board if the squares are not from the same row or column is equal to
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Let \(P_1, P_2 \ldots, P_{15}\) be 15 points on a circle. The number of distinct triangles formed by points \(P_i, P_j, P_k\) such that \(i+j+k \neq 15\), is:
[JEE Main 2021, 1 Sep (Shift 2)]
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The number of ordered pairs \( (r, k) \) for which \( 6 \cdot { }^{35} C_{r}= \) \( \left(k^{2}-3\right) \cdot { }^{36} C_{r+1} \), where \( k \) is is an integer, is
[JEE Main 2020, 7 Jan (Shift 2)]
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A scientific committee is to formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed is:
[JEE Main 2021, 24 Feb (Shift 1)]
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Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?
[JEE Main 2020, 6 Sep (Shift 1)]
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Team A consists of 7 boys and \( n \) girls and Team B has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then \( n \) is equal to :
[JEE Main 2021, 17 Mar (Shift 1)]
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Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840 , then the total numbers of persons, who participated in the tournament, is________
[JEE Main 2023, 10 Apr (Shift 1)]
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Number of words from the letters of the words BHARAT in which B and \(H\) will never come together is
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How many numbers lying between 999 and 10000 can be formed with the help of the digits \(\ 0,2,3,6,7,8 \), when the digits are not repeated?
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Team ' \(A\) ' consists of 7 boys and \(n\) girls and Team ' \(B\) ' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then \(n\) is equal to:
[JEE Main 2021, 17 Mar (Shift 1)]
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A person invites a party of 10 friends at dinner and place them so that 4 are on one round table and 6 on the other round table. The number of ways in which he can arrange the guests is
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Fractional part of the number \(\frac{4^{2022}}{15}\) is equal to
[JEE Main 2023, 13 Apr (Shift 1)]
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The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:
[JEE Main 2023, 29 Jan (Shift 2)]
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All the arrangements, with or without meaning, of the word \(\mathrm{FARMER}\) are written excluding any word that has two \(\mathrm{R}\) appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word \(\mathrm{FARMER}\) in this list is___
[JEE Main 2021, 1 Sep (Shift 2)]
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All words, with or without meaning, are made using all the letters of the word \(\mathrm{MONDAY.}\) These words are written as in a dictionary with serial numbers. The serial number of the word \(\mathrm{MONDAY}\) is
[JEE Main 2023, 13 Apr (Shift 2)]
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Total number of 6-digit numbers in which only one digit is repeated and all the five digits 1, 3, 5, 7 and 9 appear, is
[JEE Main 2020, 7 Jan (Shift 1)]
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The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits 0,1 , \(3,5,7\) and 9 without repetition, is equal to
[JEE Main 2023, 12 Apr (Shift 1)]
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The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number is
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The sum of all 3-digit numbers less then or equal to 500 , that are formed without using the digit ' 1 ' and they all are multiple of 11 , is_____
[JEE Main 2021, 26 Aug (Shift 2)]
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If \( a, b \) and \( c \) are the greatest value of \( { }^{19} C_{p},{ }^{20} C_{q} \) and \( { }^{21} C_{r} \) respectively, then
[JEE Main 2020, 8 Jan (Shift 1)]
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The largest natural number \(n\) such that \({3}^{n}\)divides 66 ! is_______
[JEE Main 2023, 8 Apr (Shift 1)]
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A natural number has prime factorization given by \( n=2^{x} 3^{y} 5^{z} \), where \( y \) and \( z \) are such that \( y+z=5 \) and \( y^{-1}+z^{-1}=\frac{5}{6}, y>z \). Then the number of odd divisors of, including 1 , is:
[JEE Main 2021, 26 Feb (Shift 2)]
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The number of six letter words (with or without meaning), formed using all the letters of the word '\(\mathrm{VOWELS}\)', so that all the consonants never come together, is
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Let S = {1, 2, 3,...9}. For k =1, 2,…5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then
N1 + N2 + N3 + N4 + N5 =
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Let the number \((22)^{2022}+(2022)^{22}\) leave the remainder \(\alpha\) when divided by 3 and \(\beta\) when divided by 7 . Then \(\left(\alpha^2+\beta^2\right)\) is equal to
[JEE Main 2023, 10 Apr (Shift 2)]
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The total number of three-digit numbers, divisible by \(3,\) which can be formed using the digits \(1,3,5,8\), if repetition of digits is allowed, is:
[JEE Main 2023, 15 Apr (Shift 1)]
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The number of seven digits odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is_______
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If the number of five digit numbers with distinct digits and \(2\) at the \( 10^{\text {th }} \) place is \( 336 \mathrm{k} \), then \( \mathrm{k} \) is equal to :
[JEE Main 2020, 9 Jan (Shift 1)]
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The number of 4-digit numbers which are neither multiple of \(7\) nor multiple of \(3\) is
[JEE Main 2021, 31 Aug (Shift 2)]
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In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_______
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All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is
[JEE Main 2023, 6 Apr (Shift 2)]
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An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during \(1-15\) June 2021 is_________
[JEE Advanced 2020]
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The number of ways of selecting two numbers \(a\) and \(b, a\) \(\in\{2,4,6, \ldots, 100\}\) and \(b \in\{1,3,5, \ldots .99\}\) such that 2 is the remainder when \(a+b\) is divided by 23 is
[JEE Main 2023, 30 Jan (Shift 2)]
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The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :
[JEE Main 2021, 18 Mar (Shift 1)]
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The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is
[JEE Main 2023, 29 Jan (Shift 2)]
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The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is
[JEE Main 2023, 8 Apr (Shift 1)]
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If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which \(C\) and \(S\) do not come together, is \((6 !) k\), then \(k\) is equal to
[JEE Main 2023, 8 Apr (Shift 2)]
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The number of possible outcomes in a throw of \( n \) ordinary dice in which at least one of the dice shows an odd number is.
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The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only is
[JEE Main 2021, 26 Feb (Shift 1)]
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The number of ways of giving 20 distinct oranges to 3 children such that each child gets atleast one orange is_______
[JEE Main 2023, 6 Apr (Shift 1)]
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A group of students comprises of \(5\) boys and \(n\) girls. If the number of ways, in which a team of \(3\) students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is \(1750\) , then \(n\) is equal to :
[JEE Main 2019, 12 Apr (Shift 2)]
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There are \(5\) students in class \(10,6\) students in class \(11\) and \(8\) students in class \(12\). If the number of ways, in which \(10\) students can be selected from them so as to include at least \(2\) students from each class and at most \(5\) students from the total \(11\) students of class \(10\) and \(11\) is \( 100 \mathrm{k} \), then \( k \) is equal to
[JEE Main 2021, 25 Jul (Shift 1)]
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The total number of three-digit numbers, divisible by 3 , which can be formed using the digits \(1,3,5,8\), if repetition of digits is allowed, is:
[JEE Main 2023, 15 Apr (Shift 1)]
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In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is_______
[JEE Main 2023, 11 Apr (Shift 1)]
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If 16 identical pencils are distributed among 4 children such that each gets at least 3 pencils. The number of ways of distributing the pencils is
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The number of 6 -digit numbers that can be made using all the digits \(0,1,2,3,4\) and 5 so that even digits occupy odd places, is
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If \({ }^n P_r={ }^n P_{r+1}\) and \({ }^n C_r={ }^n C_{r-1}\), then the value of \(r\) is equal to:
[JEE Main 2021, 25 Jul (Shift 2)]
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The total number of three-digit numbers, divisible by 3, which can be formed using the digits 1,3,5,8, if repetition of digits is allowed, is:
[JEE Main 2023, 15 Apr (Shift 1)]
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The number of ways of selecting two numbers \(a\) and \(b, a\) \(\in\{2,4,6, \ldots, 100\}\) and \(b \in\{1,3,5, \ldots .99\}\) such that \(2\) is the remainder when \(a+b\) is divided by \(23\) is
[JEE Main 2023, 30 Jan (Shift 2)]
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The number of six letter words (with or without meaning), formed using all the letters of the word '\(\mathrm{VOWELS}\)', so that all the consonants never come together, is
[JEE Main 2021, 31 Aug (Shift 1)]
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If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which \(C\) and \(S\) do not come together, is \((6 !) k\), then \(k\) is equal to
[JEE Main 2023, 08 Apr (Shift 2)]
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Among the statements:
(S1): 20232022 – 19992022 is divisible by 8.
(S2): 13(13)n – 11n – 13 is divisible by 144 for infinitely many n ∈ N.
[JEE Main 2023, 6 Apr (Shift 2)]
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If \({}^{2n}C_{3}:{}^{n}C_{3}=10:1\), then the ratio \(\left({n}^{2}+3n\right):\left({n}^{2}-3n+4\right)\) is
[JEE Main 2023, 6 Apr (Shift 1)]
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The number of integers, greater than 7000 that can be formed, using the digits \(3,5,6,7,8\) without repetition, is
[JEE Main 2023, 24 Jan (Shift 2)]
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