Permutations and Combinations · Mathematics · JEE Advanced
MCQ (Single Correct Answer)
1
The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
IIT-JEE 2007
2
Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?
JEE Advanced 2022 Paper 2 Online
3
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.
(i) Let $$\alpha $$1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $$\alpha $$2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
i) Let $$\alpha $$3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $$\alpha $$4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.
The correct option is
(i) Let $$\alpha $$1 be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $$\alpha $$2 be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
i) Let $$\alpha $$3 be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $$\alpha $$4 be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls such that both M1 and G1 are NOT in the committee together.
| LIST-I | LIST-II |
|---|---|
| P. The value of $\alpha_1$ is | 1. 136 |
| Q. The value of $\alpha_2$ is | 2. 189 |
| R. The value of $\alpha_3$ is | 3. 192 |
| S. The value of $\alpha_4$ is | 4. 200 |
| 5. 381 | |
| 6. 461 |
JEE Advanced 2018 Paper 2 Offline
4
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among these 4 members ) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
JEE Advanced 2016 Paper 1 Offline
5
Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is
JEE Advanced 2014 Paper 2 Offline
6
Let $${{a_n}}$$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let $${{b_n}}$$ = the number of such n-digit integers ending with digit 1 and $${{c_n}}$$ =the number of such n-digit integers ending with digit 0.
The value of $${{b_6}}$$ is
IIT-JEE 2012 Paper 2 Offline
7
Let $${{a_n}}$$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Let $${{b_n}}$$ = the number of such n-digit integers ending with digit 1 and $${{c_n}}$$ =the number of such n-digit integers ending with digit 0.
Which of the following is correct?
IIT-JEE 2012 Paper 2 Offline
8
The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is
IIT-JEE 2012 Paper 1 Offline
9
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
IIT-JEE 2009 Paper 1 Offline
10
Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | The number of permutations containing the word ENDEA is | (P) | 5! |
| (B) | The number of permutations in which the letter E occurs in the first and the last position is | (Q) | 2 $$\times$$ 5! |
| (C) | The number of permutations in which none of the letters D, L, N occurs in the last five positions is | (R) | 7 $$\times$$ 5! |
| (D) | The number of permutations in which the letters A, E, O occur only in odd positions is | (S) | 21 $$\times$$ 5! |
IIT-JEE 2008 Paper 2 Offline
11
The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is :
IIT-JEE 2007 Paper 2 Offline
12
If the LCM of p, q is $${r^2}\,{r^4}\,{s^2}$$, where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) is
IIT-JEE 2005 Screening
13
A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is
IIT-JEE 2005 Screening
14
The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is
IIT-JEE 2002 Screening
15
Let $${T_n}$$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $${T_{n + 1}} - {T_n} = 21$$, then n equals
IIT-JEE 2001 Screening
16
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
IIT-JEE 2000 Screening
17
A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is
IIT-JEE 1989
18
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from amongst the remaining. The number of possible arrangements is
IIT-JEE 1982
19
The value of the expression $$\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}} $$ is equal to
IIT-JEE 1982
20
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at least one letter repeated are
IIT-JEE 1982
21
$${}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126$$, then r is :
IIT-JEE 1979
Numerical
1
Let $S = \{1, 2, 3, \ldots, 10\}$. Consider the set
$X = \{R : R \text{ is an equivalence relation on the set } S \text{ such that } R \text{ has exactly 42 elements}\}$.
Then the number of elements in $X$ is ____________.
JEE Advanced 2026 Paper 1 Online
2
The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ______________.
JEE Advanced 2026 Paper 1 Online
3
Let the set of all relations $R$ on the set $\{a, b, c, d, e, f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly $10$ elements, be denoted by $\mathcal{S}$.
Then the number of elements in $\mathcal{S}$ is ________________.
JEE Advanced 2025 Paper 1 Online
4
Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example, $2210222$ is in $S$, but $0210222$ is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits $0$ and $1$ appears exactly twice in $x$, is equal to ____________.
JEE Advanced 2025 Paper 1 Online
5
If $n(X)={ }^m C_6$, then the value of $m$ is _____
JEE Advanced 2024 Paper 2 Online
6
If the value of $n(Y)+n(Z)$ is $k^2$, then $|k|$ is _________.
JEE Advanced 2024 Paper 2 Online
7
A group of 9 students, $s_1, s_2, \ldots, s_9$, is to be divided to form three teams $X, Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s_1$ cannot be selected for the team $X$, and $s_2$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is ____________.
JEE Advanced 2024 Paper 1 Online
8
The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $$0,2,3,4,6,7$$ is _________.
JEE Advanced 2022 Paper 1 Online
9
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 Paper 2 Offline
10
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 ..........
JEE Advanced 2020 Paper 2 Offline
11
Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 $$ \le $$ |B| < |A|, equals .............
JEE Advanced 2019 Paper 2 Offline
12
Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ............
JEE Advanced 2019 Paper 2 Offline
13
The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is .................
JEE Advanced 2018 Paper 1 Offline
14
Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $${y \over {9x}}$$ = ?
JEE Advanced 2017 Paper 1 Offline
15
Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of $${m \over n}$$ is
JEE Advanced 2015 Paper 1 Offline
16
Let $${n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}$$ be positive integers such that $${n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_5}$$ = 20. Then the number of such destinct arrangements $$\,({n_1}\,,\,{n_2},\,\,{n_3},\,\,{n_4}\,,{n_5})$$ is
JEE Advanced 2014 Paper 1 Offline
17
Let $${n \ge 2}$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is
JEE Advanced 2014 Paper 1 Offline
18
Consider the set of eight vectors $$V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}$$. Three non-coplanar vectors can be chosen from v in $${2^p}$$ ways. Then p is
JEE Advanced 2013 Paper 1 Offline
19
Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation:
$$$\matrix{
{3x - y - z = 0} \cr
{ - 3x + z = 0} \cr
{ - 3x + 2y + z = 0} \cr
} $$$
Then the number of such points for which $$x^2 + {y^2} + {z^2} \le 100$$ is
IIT-JEE 2009 Paper 2 Offline
MCQ (More than One Correct Answer)
1
Let
$${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,
$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$$,
$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$$ and
$${S_4} = \{ (i,j,k,l):i,j,k$$ and $$l$$ are distinct elements in {1, 2, ...., 10}.
If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?
$${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,
$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}$$,
$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \right\}$$ and
$${S_4} = \{ (i,j,k,l):i,j,k$$ and $$l$$ are distinct elements in {1, 2, ...., 10}.
If the total number of elements in the set Sr is nr, r = 1, 2, 3, 4, then which of the following statements is(are) TRUE?
JEE Advanced 2021 Paper 2 Online
2
An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is
IIT-JEE 1998
Subjective
1
If total number of runs scored in n matches is $$\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,$$ where $$n > 1$$, and the runs scored in the $${k^{th}}$$ match are given by k. $$\,{2^{n + 1 - k}}$$, where $$1 \le k \le n$$. Find n.
IIT-JEE 2005
2
Prove by permulation or otherwise $${{({n^2})!} \over {{{(n!)}^n}}}$$ is an integer $$(n \in {1^ + })$$.
IIT-JEE 2004
3
A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to included in a committee? In how many of these committees? In how may of these committees
(a) The women are in majority?
(b) The men are in majority?
(a) The women are in majority?
(b) The men are in majority?
IIT-JEE 1994
4
Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made.
IIT-JEE 1991
5
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
IIT-JEE 1986
6
7 relatives of a man comprises 4 ladies and 3 gentlemen ; his wife has also 7 relatives ; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relatives and 3 of the wife's relatives?
IIT-JEE 1985
7
m men and n women are to be seated in a row so that no two women sit together. If $$m > n$$, then show that the number of ways in which they can be seated is $$\,{{m!(m + 1)!} \over {(m - n + 1)!}}$$
IIT-JEE 1983
8
Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?
IIT-JEE 1981
9
Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.
IIT-JEE 1978
Fill in the Blanks
1
There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is.......................
IIT-JEE 1988
2
Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................
IIT-JEE 1988
3
The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .......................
IIT-JEE 1984
4
In a certain test, $${a_i}$$ students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. The total number of wrong answers given is.....................................
IIT-JEE 1982