If $\frac{1}{8!} + \frac{1}{9!} = \frac{x}{10!},$ find $x.$

In how many ways can $5$ distinct balls be distributed among $3$ persons such that each person receives at least one ball?

How many six-digit numbers are there in which no digit is repeated,even digits appear at even places,odd digits appear at odd places,and the number is divisible by $4$?

The number of all four-digit numbers which do not have four distinct digits is

What is the total number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are the four cards of the same suit?

$(2n + 1) (2n + 3) (2n + 5) \dots (4n - 1)$ is equal to :