Math Notes

Pascal's rule, a property of binomials:

\(\binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}\); where \(n,k \in \N\) and \(n>0, k>0\)

Proof:

\( \begin{aligned} \binom{n-1}{k}+\binom{n-1}{k-1} &= \\[1.2em] &= \frac{(n-1)!}{k!(n-k-1)!}+\frac{(n-1)!}{(k-1))!(n-1-k+1)!}= \\[1.2em] &= (n-1)!\frac{(n-k)+k}{k!(n-k)!}= \\[1.2em] &= (n-1)!\frac{n}{k!(n-k)!}= \\[1.2em] &= \frac{n!}{k!(n-k)!}= \\[1.2em] &= \binom{n}{k}\end{aligned} \)