Mathematical Reasoning — Maths STD 11 Science — Question

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

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 $m, n$