A certain question paper contains three parts $A, B, C$ with four questions in part $A$, five questions in part $B$ and six questions in part $C$. A student is required to answer seven questions choosing at least two questions from each part. Then the total number of different ways a student can choose his seven questions for answering, is

For $n=1,2,3, \ldots .50$, let

$$ A=\left\{a_n / a_n=\left\{\begin{array}{ll} (-1)^{\frac{n}{2}}\left(\frac{n}{2}\right), & \text { if } n \text { is even } \\ (-1)^{\frac{n-1}{2}}\left(\frac{n-1}{2}\right), & \text { if } n \text { is odd } \end{array}\right\}\right\} $$

and $B$ is the set of all distinct elements of $A$. The number of permutations all the elements of set $B$ such that even integers are in increasing order, is

If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition, then $\alpha: \beta$ is

$n^5-5 n^3+4 n$ is divisible by 120 is true for