Show that C (n+1, k) = C (n, k -1) + C (n, k)

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$C (n + 1, k) = (k n + 1) = \frac{( n + 1 )!}{k ! ( n + 1 - k )!}$

$C(n,k-1)=\dbinom{n}{k-1}=\dfrac{n!}{(k-1)!(n-k+1)!}$

$C(n,k)=\dbinom{n}{k}=\dfrac{n!}{k!(n-k)!}$

Then

$C(n,k-1)+C(n,k)$

$=\dfrac{n!}{(k-1)!(n-k+1)!}+\dfrac{n!}{k!(n-k)!}$

$=\dfrac{n!(k+n-k+1)}{k!(n-k+1)!}=\dfrac{n!(n+1)}{k!(n-k+1)!}$

$=\dfrac{(n+1)!}{k!(n+1-k)!}=C(n+1,k)$