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 $..........$.

There are $10$ persons named $A, B, \dots, J$. We have space to keep only $5$ of them. If $A$ must be included,and $G$ and $H$ must not be included in the group of $5$,in how many ways can we arrange the group in a row?

$A$ die is rolled three times. Find the probability of getting a larger number than the previous number each time.

How many words of $4$ distinct letters can be formed using the letters of the word $DHOLPUR$ such that $L$ and $P$ are always included?

If ${ }^{n} C_{12}={ }^{n} C_{8}$,then $n$ is equal to

The number of ways in which $16$ identical oranges can be distributed among $4$ children such that each child gets at least one orange is: