In a conference of $8$ persons,if each person shakes hands with every other person exactly once,then the total number of handshakes is:

An examiner can assign $30$ marks to $8$ questions. If he does not assign less than $2$ marks to any question,in how many ways can he assign the marks?

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples,is

$^{20}C_1 + 3 ^{20}C_2 + 3 ^{20}C_3 + ^{20}C_4$ is equal to-

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

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