Permutations and Combinations — JEE Main PYQs

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 :

The number of functions $f : \{1, 2, 3, 4\} \to \{a, b, c\}$, which are not onto, is:

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:

Let

$$A = \{(a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22\}.$$

Then $n(A)$ is equal to:

Let $A = \{1, 2, 3, 4, 5, 6\}$. The number of one-one functions $f: A \to A$ such that $f(1) \geq 3$, $f(3) \leq 4$ and $f(2) + f(3) = 5$, is __________.

Two players A and B play a series of games of badminton. The player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player A wins the series is ________.

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 :

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 INCONSEQUENTIAL, without repeating any letter, is:

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 $10^{\text{th}}$ 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 :

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 :

The largest $n \in \mathbf{N}$ such that $3^n$ divides $50!$ is :

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to :

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

[Figure: an arrangement of 8 boxes in three rows — top row of 3 boxes, middle row of 3 boxes, bottom row of 2 boxes, with the rows offset from one another]

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^n$; $m, n \in \{1, 2, 3, \ldots, 9\}$, then $m + n$ is equal to __________

Line $\mathrm{L}_1$ of slope 2 and line $\mathrm{L}_2$ of slope $\dfrac{1}{2}$ intersect at the origin O. In the first quadrant, $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{12}$ are 12 points on line $\mathrm{L}_1$ and $\mathrm{Q}_1, \mathrm{Q}_2, \ldots, \mathrm{Q}_9$ 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 O, $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{12}$, $\mathrm{Q}_1, \mathrm{Q}_2, \ldots, \mathrm{Q}_9$, is:

Let $\mathrm{m}$ and $\mathrm{n}$, $(\mathrm{m}<\mathrm{n})$, be two 2-digit numbers. Then the total numbers of pairs $(\mathrm{m},\mathrm{n})$, such that $\gcd(\mathrm{m},\mathrm{n})=6$, is __________.

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

For $n \geq 2$, let $S_n$ denote the set of all subsets of $\{1, 2, \ldots, n\}$ with no two consecutive numbers. For example $\{1, 3, 5\} \in S_6$, but $\{1, 2, 4\} \notin S_6$. Then $n(S_5)$ is equal to ______

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 $\dfrac{x^2}{16} + \dfrac{y^2}{n} = 1$ is :

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

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :

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 :

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

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is _________.

Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where P is the set of all possible products of distinct elements of S. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that x divides y, is _____.

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is _____.

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 :

Number of functions $f : \{1, 2, \ldots, 100\} \to \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to __________.

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

Let $^{n}C_{r-1} = 28$, $^{n}C_{r} = 56$ and $^{n}C_{r+1} = 70$. Let A $(4\cos t, 4\sin t)$, B $(2\sin t, -2\cos t)$ and C $(3r - n, r^2 - n - 1)$ be the vertices of a triangle ABC, where $t$ is a parameter. If $(3x - 1)^2 + (3y)^2 = \alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals

The number of n… [remainder of stem corrupted in the source PDF; the Hindi duplicate is also lost]

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 :

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is ___________

The number of ways of getting a sum 16 on throwing a dice four times is ________.

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 $A \cap B = B \cap C = A \cap C = \phi$. The maximum number of such possible partitions of $S$ is equal to :

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text{th}}$ word is :

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

If all the words with or without meaning made using all the letters of the word “NAGPUR” are arranged as in a dictionary, then the word at $315^{\text{th}}$ position in this arrangement is :

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to __________.

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 :

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is ____________.

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 :

The number of elements in the set $S = \{(x, y, z) : x, y, z \in \mathbb{Z},\ x + 2y + 3z = 42,\ x, y, z \ge 0\}$ equals ________.

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point P. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to ________.

$^{n-1}C_r = (k^2 - 8)\ ^nC_{r+1}$ if and only if :

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is __________.

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections : $A$, $B$ and $C$. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section $A$ has 8 questions, section $B$ has 6 questions and section $C$ has 6 questions, then the total number of ways a student can select 15 questions is ______.

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _____

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is

If for some $m, n$; ${ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2}>{ }^{8} C_{3}$ and ${ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8$, then ${ }^{n} P_{m+1}+{ }^{\mathrm{n}+1} C_{m}$ is equal to