$A$ student is required to answer $10$ questions out of $13$ in an examination,such that he must choose at least $4$ questions from the first $5$ questions. How many ways can he make the selection?

If ${}^n C_7 = {}^n C_6$,then ${}^n C_2 = $

The number of ways of selecting a committee of $30$ persons from $20$ boys,$20$ girls,and $20$ teachers such that the participation of the number of boys,girls,and teachers in that committee is equal,is

There are $15$ people at a party and each person shakes hands with every other person. What is the total number of handshakes?

$A$ committee of $11$ members is to be formed from $8$ males and $5$ females. If $m$ is the number of ways the committee is formed with at least $6$ males and $n$ is the number of ways the committee is formed with at least $3$ females,then:

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