For non-negative integers s and r, let ((s),(r)) = {(s!/r!(s - r)! if r ≤ s), (0 if r > s). For positive integers m and n, let

Question

For non-negative integers s and r, let

\(\left( \begin{array}{c} s \\ r \end{array} \right) =\begin{cases} \frac{s!} {r!(s - r)!} & \quad \text{if } r\, ≤\,s\text{},\\ 0 & \quad \text{if } r\,>\,s \text{ } \end{cases}\)

For positive integers m and n, let

g(m,n) \( \displaystyle\sum_{p=0}^{m+n}\frac{f(m,n,p)}{\left(\! \begin{array}{c} n +p\\ p \end{array} \!\right)} \)

where for any nonnegative integer p,

f(m,n,p) = \( \displaystyle\sum_{i=0}^{p}{\left(\! \begin{array}{c} m\\ i \end{array} \!\right)} \left(\! \begin{array}{c} n+i\\ p \end{array} \!\right)\left(\! \begin{array}{c} p+n\\ p-i \end{array} \!\right)\)

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

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

Answer 1

(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,

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

Now