The number of ways of distributing $500$ dissimilar boxes equally among $50$ persons is:

The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

If $4$ red balls and $5$ green balls are selected from $n$ balls,and the sum of both selections is greater than ${}^{n+1}C_4$,then the value of $n$ is equal to:

In how many ways can a committee of $5$ members be formed from $4$ gentlemen and $6$ ladies such that the number of gentlemen is greater than the number of ladies?

Determine $n$ if $^{2n}C_{3} : ^{n}C_{3} = 12 : 1$.

If a question paper consists of $11$ questions divided into two sections,$I$ and $II$. Section $I$ consists of $6$ questions and section $II$ consists of $5$ questions,then the number of different ways a student can select $6$ questions,taking at least $2$ questions from each section,is