Permutations and Combinations

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 _____.

Let $S = \left\{A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0,1\} \text{ and } |A| \in \{-1, 1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is _____.

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 _____.

If the value of $n(Y) + n(Z)$ is $k^2$, then $|k|$ 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

Let $X$ be the set of all five digit numbers formed using 1, 2, 2, 2, 4, 4, 0. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to _____.

Let $R = \left\{\begin{pmatrix}a&3&b\\c&2&d\\0&5&0\end{pmatrix} : a, b, c, d \in \{0, 3, 5, 7, 11, 13, 17, 19\}\right\}$. Then the number of invertible matrices in $R$ is _____.

Let $p_i$ be the probability that a randomly chosen point has $i$ many friends, $i = 0, 1, 2, 3, 4$. Let $X$ be a random variable such that for $i = 0, 1, 2, 3, 4$, the probability $P(X = i) = p_i$. Then the value of $7E(X)$ 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 _____.

The number of 4-digit integers in the closed interval $[2022, 4482]$ formed by using the digits $0, 2, 3, 4, 6, 7$ 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 ______.

The number of one-one functions $f:\{a,b,c,d\}\to\{0,1,2,\dots,10\}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is ______.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is ______.

Let A be a $3\times3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices A having sum of all the entries equal to 5, is ______.

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

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 ______.

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______.