Help: Multiple summation problem

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 $2 \dbinom{ n+1 }{4}$ .

#Combinatorics

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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 $2 {n+1 \choose 4}$

A Former Brilliant Member - 5 years, 1 month ago

thank you very much

Swagat Panda - 5 years, 1 month ago

In fact, we have,

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

Prasun Biswas - 4 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$