Permutations and Combinations · Mathematics · JEE Advanced

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MCQ (Single Correct Answer)

JEE Advanced 2022 Paper 2 Online
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...
JEE Advanced 2018 Paper 2 Offline
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 th...
JEE Advanced 2016 Paper 1 Offline
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 thes...
JEE Advanced 2014 Paper 2 Offline
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 ...
IIT-JEE 2012 Paper 2 Offline
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.Le...
IIT-JEE 2012 Paper 2 Offline
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.Le...
IIT-JEE 2012 Paper 1 Offline
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 2009 Paper 1 Offline
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 2008 Paper 2 Offline
Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions ...
IIT-JEE 2007
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 ...
IIT-JEE 2005 Screening
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,...
IIT-JEE 2005 Screening
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) i...
IIT-JEE 2002 Screening
The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is
IIT-JEE 2001 Screening
Let $${T_n}$$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $${T_{n + 1}} - {T_n} = 21$$, ...
IIT-JEE 2000 Screening
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 1989
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 d...
IIT-JEE 1982
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 l...
IIT-JEE 1982
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 mar...
IIT-JEE 1982
The value of the expression $$\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}} $$ is equal to
IIT-JEE 1979
$${}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126$$, then r is :

Numerical

JEE Advanced 2022 Paper 1 Online
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 2020 Paper 2 Offline
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 p...
JEE Advanced 2020 Paper 2 Offline
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 ...
JEE Advanced 2019 Paper 2 Offline
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 an...
JEE Advanced 2019 Paper 2 Offline
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, th...
JEE Advanced 2018 Paper 1 Offline
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 2017 Paper 1 Offline
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...
JEE Advanced 2015 Paper 1 Offline
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 ...
JEE Advanced 2014 Paper 1 Offline
Let $${n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}$$ be positive integers such that $${n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_...
JEE Advanced 2014 Paper 1 Offline
Let $${n \ge 2}$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining ev...
JEE Advanced 2013 Paper 1 Offline
Consider the set of eight vectors $$V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}$$. Three non-co...
IIT-JEE 2009 Paper 2 Offline
Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$$\matrix{ {3x - y - z = 0} ...

MCQ (More than One Correct Answer)

JEE Advanced 2021 Paper 2 Online
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\} } \r...
IIT-JEE 1998
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 an...

Subjective

IIT-JEE 2005
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 scor...
IIT-JEE 2004
Prove by permulation or otherwise $${{({n^2})!} \over {{{(n!)}^n}}}$$ is an integer $$(n \in {1^ + })$$.
IIT-JEE 1994
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...
IIT-JEE 1991
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 ...
IIT-JEE 1986
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 bal...
IIT-JEE 1985
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 t...
IIT-JEE 1983
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...
IIT-JEE 1981
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...
IIT-JEE 1978
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 ...

Fill in the Blanks

IIT-JEE 1988
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
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 ...
IIT-JEE 1984
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 t...
IIT-JEE 1982
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. Th...

True or False

IIT-JEE 1985
The product of any r consecutive natural numbers is always divisible by r!
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