第1回 変なΣ 「連続3回」
$$\sum_{1\leq i < j < k \leq n} = \binom{n}{3}$$
$$\sum_{1\leq i \leq j \leq k \leq n} = \left(\binom{n}{3}\right)$$
$$ \begin{align} \sum_{1\leq i < j < k \leq n} &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} \binom{i-1}{0}\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} \left[\binom{i}{1}-\binom{i-1}{1}\right]\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1} \binom{j-1}{1} \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\left[\binom{j}{2}-\binom{j-1}{2}\right]\\ &=\sum_{k=3}^n \binom{k-1}{2}\\ &=\sum_{k=3}^n \left[\binom{k}{3}-\binom{k-1}{3}\right]\\ &= \binom{n}{3}. \end{align} $$
$$ \begin{align} \sum_{1\leq i \leq j \leq k \leq n} &=\sum_{k=1}^n\sum_{j=1}^k\sum_{i=1}^j \\ &=\sum_{k=1}^n\sum_{j=1}^k\sum_{i=1}^j \left(\binom{i}{0}\right)\\ &=\sum_{k=1}^n\sum_{j=1}^k\sum_{i=1}^j \left[\left(\binom{i}{1}\right)-\left(\binom{i-1}{1}\right)\right]\\ &=\sum_{k=1}^n\sum_{j=1}^k \left(\binom{j}{1}\right) \\ &=\sum_{k=1}^n\sum_{j=1}^k\left[\left(\binom{j}{2}\right)-\left(\binom{j-1}{2}\right)\right]\\ &=\sum_{k=1}^n \left(\binom{k}{2}\right)\\ &=\sum_{k=1}^n \left[\left(\binom{k}{3}\right)-\left(\binom{k-1}{3}\right)\right]\\ &= \left(\binom{n}{3}\right). \end{align} $$
$$ \sum_{1 \leq i < j < k \leq n} = |\{i,j,k|1\leq i < j < k \leq n\}| = \binom{n}{3}. $$
$$ \sum_{1\leq i \leq j \leq k \leq n} = |\{i,j,k|1\leq i \leq j \leq k \leq n\}| = \left(\binom{n}{3}\right). $$
$$\sum_{1\leq i < j < k \leq n}i = \binom{n+1}{4}$$
$$\sum_{1\leq i < j < k \leq n}j = 2\binom{n+1}{4}$$
$$\sum_{1\leq i < j < k \leq n}k = 3\binom{n+1}{4}$$
$$ \begin{align} \sum_{1\leq i < j < k \leq n}i &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1}i \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} i\binom{i-1}{0}\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} 1\binom{i}{1}\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} \left[\binom{i+1}{2}-\binom{i}{2}\right]\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1} \binom{j}{2} \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\left[\binom{j+1}{3}-\binom{j}{3}\right]\\ &=\sum_{k=3}^n \binom{k}{3}\\ &=\sum_{k=3}^n \left[\binom{k+1}{4}-\binom{k}{4}\right]\\ &= \binom{n+1}{4}. \end{align} $$
$$ \begin{align} \sum_{1\leq i < j < k \leq n}j &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1}j \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} j\binom{i-1}{0}\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} j\left[\binom{i}{1}-\binom{i-1}{1}\right]\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1} j\binom{j-1}{1} \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1} 2\binom{j}{2} \\ &=2\sum_{k=3}^n\sum_{j=2}^{k-1}\left[\binom{j+1}{3}-\binom{j}{3}\right]\\ &=2\sum_{k=3}^n \binom{k}{3}\\ &=2\sum_{k=3}^n \left[\binom{k+1}{4}-\binom{k}{4}\right]\\ &=2\binom{n+1}{4}. \end{align} $$
$$ \begin{align} \sum_{1\leq i < j < k \leq n}k &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} k \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} k\binom{i-1}{0}\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}\sum_{i=1}^{j-1} k\left[\binom{i}{1}-\binom{i-1}{1}\right]\\ &=\sum_{k=3}^n\sum_{j=2}^{k-1} k\binom{j-1}{1} \\ &=\sum_{k=3}^n\sum_{j=2}^{k-1}k\left[\binom{j}{2}-\binom{j-1}{2}\right]\\ &=\sum_{k=3}^n k\binom{k-1}{2}\\ &=\sum_{k=3}^n 3\binom{k}{3}\\ &=3\sum_{k=3}^n \left[\binom{k+1}{4}-\binom{k}{4}\right]\\ &=3\binom{n+1}{4}. \end{align} $$