$\text{Proof of Binomial Identity}$
$\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$
$\binom{n}{0} = 1 \text{ with } 1 \leq k \leq n$
\begin{aligned} \text{LHS} \quad \binom{n}{k} = \frac{P(n, k)}{k!} = \frac{\frac{n!}{(n-k)!}}{k!} &= \frac{n!}{(n-k)! k!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{P(n-1, k)}{k!} + \frac{P(n-1, k-1)}{(k-1)!} \\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{\frac{(n-1)!}{(n-1-k)!}}{k!} + \frac{(n-1)!}{[(n-1)-(k-1)]!(k-1)!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{(n-1)!}{(n-k-1)!k!} + \frac{(n-1)!}{(n-k)!(k-1)!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{(n-k)(n-1)!}{(n-k)(n-k-1)!k!} + \frac{k(n-1)!}{k(n-k)!(k-1)!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{(n-k)(n-1)!}{(n-k)!k!} + \frac{k(n-1)!}{(n-k)!k!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{(n-1)!(n-k+k)}{(n-k)!k!}\\
\text{RHS} \quad \binom{n-1}{k} + \binom{n-1}{k-1} &= \frac{n!}{(n-k)!k!} \nonumber \\
\text{LHS} &= \text{RHS} \quad \square \end{aligned}