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Q. Let $\displaystyle\sum_{ r =0}^{100} \displaystyle\sum_{ s =0}^{100}\left( C _{ r }^2+ C _{ s }^2+ C _{ r } C _{ s }\right)= m \left({ }^{2 n } C _{ n }\right)+2^{ p }$ where $m , n$ and $p$ are even natural numbers and $C_r$ represents the coefficient of $x^r$ in the expansion of $(1+x)^{100}$. Find the value of $(m+n+p)$.

Binomial Theorem

Solution:

Correct answer is '502'