The number of ways in which first,second and third prizes can be given to $5$ competitors is

The number of words that do not start and end with vowels,formed using all the letters of the word $UNIVERSITY$ such that all vowels are in alphabetical order,is

There are $8$ students appearing in an examination of which $3$ have to appear in a Mathematics paper and the remaining $5$ in different subjects. In how many ways can they be made to sit in a row if the candidates in Mathematics cannot sit next to each other?

$A$ question paper is divided into two parts $A$ and $B$,and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions by selecting at least two questions from each part is:

$A$ 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:

The exponent of $3$ in $100!$ is