For nonnegative integers s and r, let (\table s;r) =\{ \table {\frac{s!}{r!(s-r)!}} ,\if r\leqs{,};0, \if r>s{,} For positive integers m and n, let g(m, n)=\sum_{p=0} ^{m+n} {f(m, n, p)}/{ (\table {n+p};p)} where for any nonnegative integer p, f(m, n, p)=\sum_{i=0} ^{p} (\table m;i)(\table{n+i};P) (\table {p+n};{p-i}) Then which of the following statements is/are TRUE? [JEE Adv 2020 P2]

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

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

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

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

Mathematical Induction and Binomial Theorem

Section: Mathematics

Question : 3 of 8

Marks: +1, -0

For nonnegative integers s and r, let
(

) ={

r!(s−r)!

if‌r≤s,

if‌r>s,

For positive integers m and n, let
g(m,n)=

m+n

∑

p=0

f(m,n,p)

(

n+p

)

where for any nonnegative integer p,
f(m,n,p)=

∑

i=0

(

)(

n+i

) (

p+n

p−i

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

[JEE Adv 2020 P2]

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