Combinations

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 ________.

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 ________.

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 :

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 ________.

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 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 :

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 _____.

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 _____.

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 _____.

If $n(X) = {}^{m}C_6$, then the value of $m$ is _____.

Let $X = \left\{(x, y) \in \mathbb{Z}\times\mathbb{Z} : \dfrac{x^2}{8} + \dfrac{y^2}{20} < 1 \text{ and } y^2 < 5x\right\}$. Three distinct points $P$, $Q$ and $R$ are randomly chosen from $X$. Then the probability that $P$, $Q$ and $R$ form a triangle whose area is a positive integer, is

Two distinct points are chosen randomly out of the points $A_1, A_2, \ldots, A_{49}$. Let $p$ be the probability that they are friends. Then the value of $7p$ is _____.

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?

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and $n$ white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is $\dfrac6{11}$, then $n$ is equal to ______.

There are ten boys $B_1,B_2,\dots,B_{10}$ and five girls $G_1,G_2,\dots,G_5$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B_1$ and $B_2$ together should not be the members of a group, is ______.

Five numbers $x_1,x_2,x_3,x_4,x_5$ are randomly selected from the numbers $1,2,3,\dots,18$ and are arranged in the increasing order $(x_1<x_2<x_3<x_4<x_5)$. The probability that $x_2=7$ and $x_4=11$ is :

Let $S=\{E_1,E_2,\dots,E_8\}$ be a sample space of a random experiment such that $P(E_n)=\dfrac n{36}$ for every $n=1,2,\dots,8$. Then the number of elements in the set $\left\{A\subseteq S:P(A)\ge\dfrac45\right\}$ is ______.

If the constant term in the expansion of $\left(3x^3-2x^2+\dfrac5{x^5}\right)^{10}$ is $2^k\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal to :