Combinations

The number of bijective functions $f:\{1,3,5,7,\dots,99\}\to\{2,4,6,8,\dots,100\}$, such that $f(3)\ge f(9)\ge f(15)\ge f(21)\ge\dots\ge f(99)$, is ______

Let $A=\{1,2,3,4,5,6,7\}$. Define $B=\{T\subseteq A:\text{ either }1\notin T\text{ or }2\in T\}$ and $C=\{T\subseteq A:T\text{ the sum of all the elements of }T\text{ is a prime number}\}$. Then the number of elements in the set $B\cup C$ is ______.

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 ______.

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 _______.

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 ______

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 ______

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 ______

In a high school, a committee has to be formed from a group of 6 boys $M_1, M_2, M_3, M_4, M_5, M_6$ and 5 girls $G_1, G_2, G_3, G_4, G_5$.

(i) Let $\alpha_1$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.

(ii) Let $\alpha_2$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.

(iii) Let $\alpha_3$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.

(iv) Let $\alpha_4$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M_1$ and $G_1$ are NOT in the committee together.

The correct option is:

How many $3 \times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there, for which the sum of the diagonal entries of $M^TM$ is 5?

Let $S = \{1, 2, 3, \ldots, 9\}$. For $k = 1, 2, \ldots, 5$, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1 + N_2 + N_3 + N_4 + N_5 =$

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is :

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is :

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

If the number of terms in the expansion of $\left(1-\dfrac2x+\dfrac4{x^2}\right)^n$, $x\ne0$, is 28, then the sum of the coefficients of all the terms in this expansion, is :