Find the value of $$\mathop{\sum\sum\sum\sum}_{0\leq i<j<k<i\leq n} 2$$
Had it been $\displaystyle\mathop{\sum\sum\sum\sum}_{0\leq i<j<k<l\leq n} 2$, it would be $2 {{n+1}\choose 4}$.
Is the question wrong? If not, please explain.
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1See https://math.stackexchange.com/questions/2342139/finding-sum-mathop-sum-sum-sum-sum-0-le-i-lt-j-lt-k-lt-l-le-n-1/2342142 – Angina Seng Oct 28 '17 at 04:43
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1Just like your other (very similar) question, this is a straightforward consequence of stars and bars (https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)). – Jack D'Aurizio Oct 28 '17 at 16:01
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1Possible duplicate of Find the value of $\mathop{\sum\sum\sum\sum}_{0\leq i \leq j \leq k \leq l\leq n} 1$ – Simply Beautiful Art Oct 30 '17 at 14:54
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@SimplyBeautifulArt It's not a duplicate. In one case, I asked for the explanation of the question while in the other case, I asked for a solution using bijection of some form – Mathejunior Oct 30 '17 at 15:50
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Oh, okay. Coolios. – Simply Beautiful Art Oct 30 '17 at 15:56
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It is almost certainly a typo and your interpretation is correct. If it's not a typo then (a) the sum is equal to zero, since $0 \le i<j<k<i \le n$ is true for no values of $i$; and (b) there are four $\sum$ symbols but only three variables being (implicitly) summed over, which is somewhat unlikely.
Clive Newstead
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