If $20$ identical books are to be distributed among $4$ persons such that each person receives at least one book,in how many ways can this be done?

$A$ student has to answer $10$ questions,choosing at least $4$ from each of the parts $A$ and $B$. If there are $6$ questions in part $A$ and $7$ in part $B$,then the number of ways the student can choose $10$ questions is:

$A$ candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$,each containing $6$ questions. The candidate is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make his/her choice of $6$ questions?

There are ten boys $B_{1}, B_{2}, \ldots, B_{10}$ and five girls $G_{1}, G_{2}, \ldots, G_{5}$ in a class. The number of ways of forming a group consisting of three boys and three girls,such that $B_{1}$ and $B_{2}$ are not both members of the same group,is

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and two different consonants can be formed from the alphabet?

$A$ 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?