If $_n{P_4} = 24 \times \binom{n}{5}$,then $n = \dots$

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

The number of seven-digit numbers that can be formed by using the digits $1, 2, 3, 5,$ and $7$ such that each digit is used at least once is:

From a collection of $20$ consecutive natural numbers, four are selected such that they are not consecutive. The number of such selections is

The total number of seven-digit numbers whose sum of digits is even is

Let $a_1, a_2, \ldots, a_n$ be $n$ non-zero real numbers,of which $p$ are positive and the remaining are negative. The number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is positive is $55$. Similarly,the number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is negative is $50$. Then,the value of $p^2 + (n-p)^2$ is