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$\large \sum_{i = 2}^{25}P(i,2)$ $P$ stands for "permutations".

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    Yes,$ P(i,2) = {i! \over (i - 2)!}$ Second involves the sum of an arithmetic sequence.${n(n + 1) \over 2}$Third:${n(n + 1)(2n + 1)\over 6} $2012-09-16

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$\displaystyle\sum_{i=2}^{25} P(i,2) = \displaystyle\sum_{i=2}^{25} \frac{i!}{(i-2)!} = \displaystyle\sum_{i=2}^{25} \frac{i (i-1) (i-2)!}{(i-2)!} = \displaystyle\sum_{i=2}^{25} i (i-1) = \displaystyle\sum_{i=2}^{25} (i^2 - i) = \displaystyle\sum_{i=2}^{25} i^2 - \displaystyle\sum_{i=2}^{25} i$

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    @Limitless: You're right.2012-09-24
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Hint:
$i(i-1)=\frac{1}{3}\Big((i+1)(i)(i-1)-(i)(i-1)(i-2)\Big).$ Add up from $i=2$ to $i=25$, and observe the beautiful cancellations (telescoping).

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$\sum_{i=2}^{25}P(i,2)=\sum_{i=2}^{25}\frac{i!}{(i-2)!}=\sum_{i=2}^{25}i(i-1)=\sum_{i=2}^{25}i^2-\sum_{i=2}^{25}i$

There are well-known formulas for $\sum_{i=1}^ni$ and $\sum_{i=1}^ni^2$ that you can use to finish the job; these formulas can be found (among many other places) in most standard calculus texts when summations are introduced preparatory to doing Riemann sums.