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{\displaystyle {n \choose k}+{n \choose k-1}={n+1 \choose k}.}
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{\displaystyle {\begin{aligned}{n \choose k}+{n \choose k-1}&={\frac {n!}{k!(n-k)!}}+{\frac {n!}{(k-1)!(n-k+1)!}}\\&=n!\left[{\frac {n-k+1}{k!(n-k+1)!}}+{\frac {k}{k!(n-k+1)!}}\right]\\&={\frac {n!(n+1)}{k!(n-k+1)!}}={\binom {n+1}{k}}\end{aligned}}}
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{\displaystyle n,k_{1},k_{2},k_{3},\dots ,k_{p},p\in \mathbb {N} ^{*}}
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{\displaystyle n=k_{1}+k_{2}+k_{3}+\cdots +k_{p}}
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{\displaystyle {\begin{aligned}&{}\quad {n-1 \choose k_{1}-1,k_{2},k_{3},\dots ,k_{p}}+{n-1 \choose k_{1},k_{2}-1,k_{3},\dots ,k_{p}}+\cdots +{n-1 \choose k_{1},k_{2},k_{3},\dots ,k_{p}-1}\\&={\frac {(n-1)!}{(k_{1}-1)!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {(n-1)!}{k_{1}!(k_{2}-1)!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots (k_{p}-1)!}}\\&={\frac {k_{1}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {k_{2}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {k_{p}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {(k_{1}+k_{2}+\cdots +k_{p})(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}\\&={\frac {n(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {n!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={n \choose k_{1},k_{2},k_{3},\dots ,k_{p}}.\end{aligned}}}