There are $10$ persons named $A, B, \dots, J$. We have the capacity to accommodate only $5$. In how many ways can we arrange them in a line if $A$ must be included and $G$ and $H$ must not be included in the team of $5$?

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first $2$ questions have $3$ choices each,the next $2$ have $4$ choices each,and the last $2$ have $5$ choices each?

How many words can be made from the letters of the word $BHARAT$ in which $B$ and $H$ never come together?

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by $66$,then the number of men who participated in the tournament lies in the interval

In how many ways can $5$ sportsmen be selected from a group of $10$?

The value of $^n{P_r}$ is equal to