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# Help: Multiple summation problem

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

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

Note by Swagat Panda
1 year, 8 months ago

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The 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 $2 {n+1 \choose 4}$

- 1 year, 8 months ago

thank you very much

- 1 year, 8 months ago

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}

- 1 year, 3 months ago