If $^n{P_3}{ + ^n}{C_{n - 2}} = 14n$, then $n = $

If $P(n,r) = 1680$ and $C(n,r) = 70$, then $69n + r! = $

$n$ balls each of weight $w$ when weighted in pairs the sum of the weights of all the possible pairs is $120$ when they are weighed in triplets the sum of the weights comes out to be $480$ for all possible triplets, then $n$ is

If all the six digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the $72^{\text {th }}$ number is $............$.

A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to

If $'n'$ objects are arranged in a row then the number of ways of selecting three of these objects so that no two of them are next to each othe