Permutations and Combinations

Let S be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5, then $9p$ is equal to

Let S be the set containing all $3\times3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{\mathrm T}A$ is 6 is ______.

Consider a matrix $A=\begin{bmatrix}\alpha&\beta&\gamma\\\alpha^2&\beta^2&\gamma^2\\\beta+\gamma&\gamma+\alpha&\alpha+\beta\end{bmatrix}$, where $\alpha,\beta,\gamma$ are three distinct natural numbers.

If $\dfrac{\det(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32}\times3^{16}$, then the number of such 3-tuples $(\alpha,\beta,\gamma)$ is ______.

The number of functions $f$, from the set $A=\left\{x\in\mathbf N:x^2-10x+9\le0\right\}$ to the set $B=\left\{n^2:n\in\mathbf N\right\}$ such that $f(x)\le(x-3)^2+1$, for every $x\in A$, is ______.

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from {A, B, C, D, E} or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $\alpha\times5^6$, then $\alpha$ is equal to ______.

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then b + 3 g is equal to ______.

The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

$\displaystyle\sum_{r=1}^{20}(r^2+1)(r!)$ is equal to

The number of natural numbers lying between 1012 and 23421 that can be formed using the digits 2, 3, 4, 5, 6 (repetition of digits is not allowed) and divisible by 55 is _______.

The sum of all values of n, for which ${}^{n-1}C_4-{}^{n-1}C_3-\dfrac54\,{}^{n-2}P_2<0$, is

Let
$$\begin{aligned}S_1 &= \{(i, j, k) : i, j, k \in \{1, 2, \ldots, 10\}\},\\ S_2 &= \{(i, j) : 1 \le i < j + 2 \le 10,\ i, j \in \{1, 2, \ldots, 10\}\},\\ S_3 &= \{(i, j, k, l) : 1 \le i < j < k < l,\ i, j, k, l \in \{1, 2, \ldots, 10\}\}\end{aligned}$$
and
$$S_4 = \{(i, j, k, l) : i, j, k \text{ and } l \text{ are distinct elements in } \{1, 2, \ldots, 10\}\}.$$
If the total number of elements in the set $S_r$ is $n_r$, $r = 1, 2, 3, 4$, then which of the following statements is (are) TRUE?

A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M$^T$M is seven, is _______.

For nonnegative integers $s$ and $r$, let

$$\binom sr = \begin{cases}\dfrac{s!}{r!\,(s-r)!} & \text{if } r \le s,\\ 0 & \text{if } r > s.\end{cases}$$

For positive integers $m$ and $n$, let

$$g(m,n) = \sum_{p=0}^{m+n}\frac{f(m,n,p)}{\binom{n+p}{p}}$$

where for any nonnegative integer $p$,

$$f(m,n,p) = \sum_{i=0}^{p}\binom mi\binom{n+i}{p}\binom{p+n}{p-i}.$$

Then which of the following statements is/are TRUE?

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 place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is ______

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 one person and at most two persons. Then the number of all possible ways in which this can be done is ______

Let $S$ be the sample space of all $3\times3$ matrices with entries from the set $\{0, 1\}$. Let the events $E_1$ and $E_2$ be given by

$E_1 = \{A\in S : \det A = 0\}$ and

$E_2 = \{A\in S : \text{sum of entries of } A \text{ is } 7\}$.

If a matrix is chosen at random from $S$, then the conditional probability $P(E_1|E_2)$ equals ______

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, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ______

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 $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \le |B| < |A|$, equals ______

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 _____ .