The number of matrices of order $3 \times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is$….$

For non-negative integers $s$ and $r$, let

$\binom{s}{r}=\left\{\begin{array}{ll}\frac{s!}{r!(s-r)!} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{array}\right.$

For positive integers $m$ and $n$, let

$(m, n) \sum_{ p =0}^{ m + n } \frac{ f ( m , n , p )}{\binom{ n + p }{ p }}$

where for any nonnegative integer $p$,

$f(m, n, p)=\sum_{i=0}^{ p }\binom{m}{i}\binom{n+i}{p}\binom{p+n}{p-i}$

Then which of the following statements is/are $TRUE$?

$(A)$ $(m, n)=g(n, m)$ for all positive integers $m, n$

$(B)$ $(m, n+1)=g(m+1, n)$ for all positive integers $m, n$

$(C)$ $(2 m, 2 n)=2 g(m, n)$ for all positive integers $m, n$

$(D)$ $(2 m, 2 n)=(g(m, n))^2$ for all positive integers $m, n$

How many $6 -$ digit numbers can be formed from the digits, $0,1,3,5,7$ and $9$ which are divisible by $10$ and no digit is repeated?

The students $S _{1}, S _{2}, \ldots \ldots, S _{10}$ are to be divided into $3$ groups $A , B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is …….. .

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