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Mathematical Induction and Binomial Theorem question

2020 · Shift 2 · Q30

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JEE Advanced
Mathematics
Mathematical Induction and Binomial Theorem
2020 · Shift 2 · Q30

Mathematical Induction and Binomial Theorem question

2020 · Shift 2 · Q30

JEE AdvancedMathematicsMathematical Induction and Binomial TheoremMultiple correct+4 / −2

For non-negative integers s and r, let

\left( {\begin{matrix} s \\ r \\ \end{matrix} } \right) = \left\{ {\begin{matrix} {{{s!} \over {r!(s - r)!}}} & {if\,r \le \,s,} \\ 0 & {if\,r\, \gt \,s} \\ \end{matrix} } \right.

For positive integers m and n, let

g(m,\,n) = \sum\limits_{p = 0}^{m + n} {{{f(m,n,p)} \over {\left( {\begin{matrix} {n + p} \\ p \\ \end{matrix} } \right)}}}

where for any non-negative integer p,

f(m,n,p) = \sum\limits_{i = 0}^p {\left( {\begin{matrix} m \\ i \\ \end{matrix} } \right)\left( {\begin{matrix} {n + i} \\ p \\ \end{matrix} } \right)\left( {\begin{matrix} {p + n} \\ {p - i} \\ \end{matrix} } \right)}

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

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