Please help in computing the following summation.

\[\large \mathop{\sum\sum\sum\sum}_{0 \leq i < j < k < l\leq n } 2 \]

The answer to this is \(2 \dbinom{ n+1 }{4} \).

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## Comments

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TopNewestThe given sum is just twice the ways of ways of picking integers \(i, j, k, l\) with the constraint that \(0\leq i<j<k<l\leq n\) which is just \[2 {n+1 \choose 4} \]

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thank you very much

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In fact, we have,

\[\large\underset{a\leq r_1\lt r_2\lt\cdots\lt r_n\leq b}{\sum\sum\cdots\sum\sum}1=\begin{cases}\begin{align*}\dbinom{b-a+1}{n}&~&\textrm{if }b\geq a\\~\\0&~&\textrm{otherwise}\end{align*}\end{cases}\]

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