Six married couples decide to form a committee of $6$ persons. The total number of ways to form the committee such that it contains no couples is equal to:

If $^{n^2 - n}C_2 = ^{n^2 - n}C_{10}$,then $n = $

The greatest integer which divides $(p+1)(p+2)(p+3) \ldots (p+q)$ for all $p \in N$ and fixed $q \in N$ is

$A$ student has to answer $10$ questions,choosing at least $4$ from each of the parts $A$ and $B$. If there are $6$ questions in part $A$ and $7$ in part $B$,then the number of ways the student can choose $10$ questions is:

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

If ${}^nC_4, {}^nC_5$ and ${}^nC_6$ are in $A.P.$, then $n$ is