From among the $36$ teachers in a school,one principal and one vice-principal are to be appointed. In how many ways can this be done?

Eleven books consisting of $5$ Mathematics,$4$ Physics,and $2$ Chemistry are placed on a shelf. The number of possible ways of arranging them on the assumption that the books of the same subject are all together is

How many different words can be formed with the letters of the word '$BHARAT$'? In how many of these are $B$ and $H$ never together?

The number of ways in which $5$ male and $2$ female members of a committee can be seated around a round table so that the two female are not seated together is

If $^{20}C_{n+2} = ^nC_{16}$,then the value of $n$ is

The solution set of $^{10}C_{x-1} > 2 \cdot ^{10}C_x$ is