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$$ \begin{align} \binom{i}{j}\binom{j}{k}&=\binom{i}{k}\binom{i-k}{j-k}\\\\ \end{align} $$

$$ \begin{align}\\\\ x^n&=\sum_{i=0}^{n}\begin{Bmatrix}n\\i\end{Bmatrix}x^{\underline{i}}\\ &=\sum_{i=0}^{n}\begin{Bmatrix}n\\i\end{Bmatrix}\binom{x}{i}i!\\\\ x^{\underline{n}}&=\sum_{i=0}^{n}(-1)^{n-i}\begin{Bmatrix}n\\i\end{Bmatrix}x^i\\ \end{align} $$

$$ \begin{align}\\\\ (a+b)^n=\sum_{i=0}^{n}a^ib^{n-i}\binom{n}{i} \end{align} $$

$$ \begin{align}\\\\ &\sum_{i=0}^{n}(-1)^i\binom{n}{i}=[n=0]\\\\ &\sum_{i=k}^{n}(-1)^i\binom{n-k}{i-k}\\ =&(-1)^k\sum_{i=0}^{n-k}(-1)^i\binom{n-k}{i}\\ =&(-1)^k[n=k] \end{align} $$




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