# Thread: Double summation - Binomial Theorem?

1. ## Double summation - Binomial Theorem?

Find the value of:
(i. nCj + j. nCi)
1≤ i < j ≤ n
NOTE: i. nCj is i multiplied by n choose j

2. Originally Posted by fardeen_gen
Find the value of:
(i. nCj + j. nCi)
1≤ i < j ≤ n
NOTE: i. nCj is i multiplied by n choose j
In that sum, each integer from 1 to n is multiplied by each binomial coefficient $\displaystyle \textstyle n\choose k$ (k going from 1 to n), except for the "diagonal" products $\displaystyle k{n\choose k}$. So the sum is equal to $\displaystyle \sum_{i=1}^ni\sum_{j=1}^n{n\choose j} - \sum_{k=1}^nk{n\choose k}$. But $\displaystyle \sum_{i=1}^ni = \tfrac12n(n+1)$, $\displaystyle \sum_{j=1}^n{n\choose j} = 2^n-1$ and $\displaystyle \sum_{k=1}^nk{n\choose k} = 2^{n-1}n$. Therefore $\displaystyle \mathop{\sum\sum}_{1\leqslant i<j\leqslant n}\left(i{n\choose j} + j{n\choose i}\right) = \tfrac12n(n+1)(2^n-1) - 2^{n-1}n = 2^{n-1}n^2 - \tfrac12n(n+1)$.

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