$^{14}C_4 + \sum_{j=1}^{4} {^{18-j}C_3}$ 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?

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. Then the total number of possible choices available to him is

Sangeeta plans a dinner party for $6$ guests. Out of $10$ friends,in how many ways can she choose the guests if two specific friends cannot attend the party together?

The number of ways in which $6$ distinct things can be distributed into $2$ boxes so that no box is empty is