The number of integral solutions to the equation $x+y+z=21$,where $x \geq 1, y \geq 3, z \geq 4$,is equal to $..........$.

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

In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men. The number of committees in which the women are in majority is

What is the number of ways in which an examiner can assign $10$ marks to $4$ questions,giving not less than $2$ marks to any question?

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