In how many ways can a group of $7$ persons be formed from $6$ boys and $4$ girls such that the boys are in the majority?

The number of ways in which one or more balls can be selected out of $10$ white,$9$ green,and $7$ blue balls is:

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

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

$A$ group of $9$ students,$s_1, s_2, \ldots, s_9$,is to be divided into three teams $X, Y$,and $Z$ of sizes $2, 3$,and $4$,respectively. Suppose that $s_1$ cannot be selected for team $X$,and $s_2$ cannot be selected for team $Y$. The number of ways to form such teams is:

The number of integers $x, y, z, w$ satisfying $x+y+z+w=25$ and $x, y, z \geq -1, w \geq 1$ is