Question

For nonnegative integers $s$ and $r$, let
$\begin{pmatrix}s \\ r\end{pmatrix}=\begin{cases}\frac{s !}{r !(s-r) !} & \text { if } r \leq s \\ 0 & \text { if } r>s\end{cases}$
For positive integers $m$ and $n$, let
$g(m, n)-\displaystyle\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\begin{pmatrix}n+p \\ p\end{pmatrix}}$
where for any nonnegative integer $p$,
$f(m, n, p)=\displaystyle\sum_{i=0}^{p}\begin{pmatrix}m \\ i\end{pmatrix}\begin{pmatrix}n+i \\ p\end{pmatrix}\begin{pmatrix}p+n \\ p-i\end{pmatrix}$
Then which of the following statements is/are TRUE?

• A. $g ( m , n )= g ( n , m )$ for all positive integers $m , n$
• B. $g ( m , n +1)= g ( m +1, n )$ for all positive integers $m , n$
• C. $g (2 m , 2 n )=2 g ( m , n )$ for all positive integers $m , n$
• D. $g (2 m , 2 n )=( g ( m , n ))^{2}$ for all positive integers $m , n$

Answer 1

Answer: (A), (B), (D)

$f ( m , n , p )=\displaystyle\sum_{ i =0}^{p}\left({ }^{m} C_{i}{ }^{n+i} C_{p}{ }^{p+n} C_{p-i}\right)$
$=\displaystyle\sum_{i=0}^{p}\left(\frac{m !}{i !(m-1) !} \cdot \frac{(n+i) !}{p !(n+i-p) !} \cdot \frac{(p+n) !}{(p-i) !(n+i) !}\right)$
$=\displaystyle\sum_{i=0}^{p}{ }^{m} C_{i} \cdot{ }^{n+p} C_{n} \cdot{ }^{n} C_{p-i}$
$={ }^{n+p} C_{n} \cdot{ }^{m+n} C_{p}$
$g\left(m_{1} n\right)=\displaystyle\sum_{p=0}^{m+n} \frac{{ }^{m+n} C_{p}{ }^{n+p} C_{n}}{{ }^{n+p} C_{p}}=\sum_{p=0}^{m+n}{ }^{m+n} C_{p}$
$=2^{m+n} .$