From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there in the committee. In how many ways can it be done?

In a city,no two persons have an identical set of teeth,and there is no person without a tooth. Also,no person has more than $32$ teeth. If we disregard the shape and size of the teeth and consider only the presence or absence of a tooth at each of the $32$ positions,then the maximum population of the city is:

If $^{n^2 - n}C_2 = ^{n^2 - n}C_{10}$,then $n = $

In how many ways can the letters of the word '$ALGEBRA$' be arranged when both $A$ do not come together?

The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ in which at least two digits are identical is

To fill $12$ vacancies,there are $25$ candidates,of which $5$ are from the scheduled caste. If $3$ of the vacancies are reserved for scheduled caste candidates while the rest are open to all,then the number of ways in which the selection can be made is: