It is required to seat $5$ men and $4$ women in a row so that the women occupy the even places. How many such arrangements are possible?

Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is

Total number of four-digit odd numbers that can be formed using $0, 1, 2, 3, 5, 7$ (repetition is allowed) is:

Words of length $10$ are formed using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated,and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then,$\frac{y}{9x} =$

If $0 < r < s < n$ and ${}^n P_r = {}^n P_s$,then find the value of $r + s$.

$(n - r + 1) \times ^nP_{r - 1} = \dots$