For a natural number $n$,$(n!)^2 > n^n$ is true if:

Three and four digit numbers are formed from the digits $1, 3, 5, 6, 8$. If $e_1$ is the number of three-digit even numbers with no digit repeated and $e_2$ is the number of four-digit even numbers with no digit repeated. Also,$O_1$ represents the number of three-digit odd numbers in which no digit is repeated and $O_2$ represents the number of four-digit odd numbers in which no digit is repeated. Then:

Out of all possible $8$-digit numbers formed using all the digits $0, 0, 1, 1, 2, 3, 4, 4$,a number is randomly selected. The probability that the selected number is odd is:

The number of four-lettered words that can be formed from the letters of the word $MAYANK$ such that both $A$'s are included but never together,is equal to:

Consider the set $A = \{1, 2, 3, \ldots, 30\}$. The number of ways in which one can choose three distinct numbers from $A$ such that the product of the chosen numbers is divisible by $9$ is:

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