For non-negative integers and , let For positive integers and , let where for any non-negative integer , Then which of the following statements is/are TRUE? [2020]

Question

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

$(\binom{s}{r}) =$ ${\begin{matrix} \frac{s!}{r! (s - r)!} & if r \leq s, \\ 0 & if r > s . \end{matrix}$

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

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

where for any non-negative 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? [2020]

Smart Achievers · Accepted Answer

(1, 2, 4)

So, options (1), (2) and (4) are true.

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