The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

In a Mathematics examination,there are $20$ questions of equal marks. The question paper is divided into three sections: $A, B$,and $C$. $A$ student is required to attempt a total of $15$ questions,taking at least $4$ questions from each section. If section $A$ has $8$ questions,section $B$ has $6$ questions,and section $C$ has $6$ questions,then the total number of ways a student can select $15$ questions is:

The number of ways to distribute $25$ identical apples among $4$ boys,such that each boy receives at least $4$ apples,is

If $\binom{a^2+a}{3} = \binom{a^2+a}{9}$,then $a = \dots$

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

If a person has $3$ coins of different denominations,the number of different sums that can be formed is