In how many ways can one girl and one boy be selected from a group of $15$ boys and $8$ girls?

If $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n! = 11! - 1$,then the maximum value of ${}^n C_r$ is

The number of ways of choosing a committee from $4$ men and $6$ women so that the committee includes at least $2$ men and exactly twice as many women as men is

How many words can be formed by taking $3$ consonants and $2$ vowels out of $5$ consonants and $4$ vowels?

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

Let $n$ and $k$ be positive integers such that $n \ge \frac{k(k + 1)}{2}$. The number of solutions $(x_1, x_2, ..., x_k)$ where $x_1 \ge 1, x_2 \ge 2, ..., x_k \ge k$ are all integers,satisfying $x_1 + x_2 + ... + x_k = n$,is