$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

If the number of subsets with $8$ elements from the set $A=\{a_1, a_2, a_3, \ldots, a_n\}$,$n \geq 8$,is five times the number of such subsets containing $a_4$,then $n=$

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men such that it contains at least $5$ women. Find the number of ways the committee can be formed such that women are in the majority and the number of ways the committee can be formed such that men are in the majority,respectively.

The number of values of $r$ satisfying $^{69}C_{3r-1} - ^{69}C_{r^2} = ^{69}C_{r^2-1} - ^{69}C_{3r}$ is:

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

$n$ balls each of weight $w$. When weighed in pairs,the sum of the weights of all possible pairs is $120$. When they are weighed in triplets,the sum of the weights of all possible triplets is $480$. Then $n$ is: