If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

If $^nC_r = ^nC_{r-1}$ and $^nP_r = ^nP_{r+1}$,then the value of $n$ is

$A$ string of letters is to be formed by using $4$ letters from all the letters of the word "$MATHEMATICS$". The number of ways this can be done such that two letters are of the same kind and the other two are of a different kind is:

There are $3$ sections in a question paper and each section contains $5$ questions. $A$ candidate has to answer a total of $5$ questions,choosing at least one question from each section. Then the number of ways,in which the candidate can choose the questions,is

Total number of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1, 2, 3, 4, 5$ with repetition is $.......$.

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