For non-negative integers s and r , let s r = array ll s! r!(s-r)! & if r s, 0 & if r>s …
Mathematics · JEE Advanced · NTA Exams — Binomial Theorem And Its Simple Applications
For non-negative integers s and r, let
\[\binom{s}{r}=\left\{\begin{array}{ll} \frac{s!}{r!(s-r)!} & \text { if } r \leq s, \\ 0 & \text { if } r>s . \end{array}\right.\]
For positive integers m and n, let
\[g(m, n)=\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}\]
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?
\[\binom{s}{r}=\left\{\begin{array}{ll} \frac{s!}{r!(s-r)!} & \text { if } r \leq s, \\ 0 & \text { if } r>s . \end{array}\right.\]
For positive integers m and n, let
\[g(m, n)=\sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}\]
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?
- 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
Answer
(A) g ( m , n ) = g ( n , m ) for all positive integers m , n
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