In how many ways can a girl and a boy be selected from a group of $15$ boys and $8$ girls?

How many words can be formed using the letters of the word $INSURANCE$,such that all vowels always come together?

How many words can be formed with the letters of the word $MATHEMATICS$ by rearranging them?

$3$ boys and $3$ girls are to be seated around a circular table. Among the boys,$X$ does not want any girl as a neighbour,and among the girls,$Y$ does not want any boy as a neighbour. How many such arrangements are possible?

If $\alpha = {^m}{C_2}$,then ${^\alpha}{C_2}$ is equal to

$A$ man $X$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are in this party,is: