Value of ∑∑0≤r<s≤n (r+s)(Cr+Cs) 2 is

S=∑0≤r ∑<s≤n (n−r+n−s)(Cn−r−Cn−s)2 =2n∑0≤r ∑<s≤n (Cr+Cs)2 −S⇒S=n[(n−1)( 2nCn)+22n]

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Value of { _{ 0 r<s n }^{ }{ { (r+s)( { C }_{ r }+{ C }_{ s

Question

Value of $\sum \sum_{0 \leq r < s \leq n} (r + s) (C_{r} + C_{s})^{2}$ is

n [(n - 1) (_{2 n} C_{n}) + 2^{2 n}]

n [(n + 1) (_{2 n} C_{n}) + 2^{2 n}]

n [2^{2 n} - n (_{2 n} C_{n})]

None of these

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Value of

\sum \sum_{0 \leq r < s \leq n} (r + s) (C_{r} + C_{s})^{2}

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S = \sum_{0 \leq r} \sum_{< s \leq n} (n - r + n - s) (C_{n - r} - C_{n - s})^{2}

= 2 n \sum_{0 \leq r} \sum_{< s \leq n} (C_{r} + C_{s})^{2} - S

\Rightarrow S = n [(n - 1) (_{2 n} C_{n}) + 2^{2 n}]

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Question Text	Value of $\sum \sum_{0 \leq r < s \leq n} (r + s) (C_{r} + C_{s})^{2}$ is
Answer Type	Text solution:1
Upvotes	150