$A$ committee of $7$ members is to be formed from $9$ boys and $4$ girls. In how many ways can this be done if the committee must consist of exactly $3$ girls?

$A$ lady gives a dinner party for $6$ guests. The number of ways in which they may be selected from among $10$ friends,if two of the friends will not attend the party together,is:

If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^nC_{r-1} + ^nC_r = $

In how many ways can one select a cricket team of $11$ from $17$ players in which only $5$ players can bowl,if each cricket team of $11$ must include exactly $4$ bowlers?

Let $A_1, A_2, \dots, A_{11}$ be players in a team with their $T$-shirts numbered $1, 2, \dots, 11$. One hundred gold coins were won by the team in the final match. These coins are to be distributed among the players such that each player $A_i$ gets at least $i+1$ coins,except for the captain and vice-captain who get at least $5$ and $3$ coins more than their $T$-shirt numbers respectively. If the captain is $A_1$ and the vice-captain is $A_2$,in how many different ways can these coins be distributed?

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2$ and $z \geq 5$ is: