In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?

The number of integers between $100$ and $1000$ such that the sum of their digits is equal to $14$ 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:

In how many ways can we select any $4$ letters from the word $CORGOO$?

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

The domain of definition of the function $f(x) = \log_{[x + \frac{1}{x}]} |x^2 - x - 6| + ^{16-x}C_{2x-1} + ^{20-3x}P_{2x-5}$ is,where $[x]$ denotes the greatest integer function.