The number of ways in which $4$ different things can be distributed to $6$ persons so that no person gets all the things is

Let $P_m$ stand for $^nP_m$. Then the expression $1 \cdot P_1 + 2 \cdot P_2 + 3 \cdot P_3 + \dots + n \cdot P_n =$

Consider $4$ boxes,where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

In a club election,the number of contestants is one more than the maximum number of candidates for which a voter can vote. If the total number of ways in which a voter can vote is $62$,then the number of candidates is:

The sum of all possible numbers that can be formed by using the digits $2, 3, 5, 7$ without repetition of digits is

The number of distinct primes dividing $12! + 13! + 14!$ is