Permutations and Combinations

Let $S=\{1,2,3,\dots,10\}$. 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 ________.

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 ________.

Let $\mathbb N$ denote the set of all positive integers. Consider the sets
$$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$$
Let $S$ be the set of all functions $f:A\to B$ such that $f(2)\ne2$ and $f(4)\ne4$. Consider the set
$$T=\left\{f\in S:\text{there exists a function }g:B\to\mathbb N\text{ such that }g(f(x))=2^x\text{ for all }x\in A\right\}.$$
Then the number of elements in the set $T$ is ________.

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $X$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $X$, then the value of $77\alpha$ is ________.

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 :

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:

Let $S=\left\{A=\begin{bmatrix}a&b\\c&d\end{bmatrix}:a,b,c,d\in\{0,1,2,3,4\}\ \text{and}\ A^2-4A+3I=0\right\}$ be a set of $2\times2$ matrices. Then the number of matrices in S, for which the sum of the diagonal elements is equal to 4, is:

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

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$, then $96p$ is equal to ________.

Let $A=\{(a,b,c):a,b,c$ 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 $f(1)\ge3$, $f(3)\le4$ 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 10th 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 bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

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 :

Consider the matrix

$$P = \begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{pmatrix}.$$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times3$ invertible matrices $Q$ with integer entries, such that

$$Q^{-1} = Q^T \quad\text{and}\quad PQ = QP,$$

is

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 _____.