Question

Prove that ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$.

Answer 1

Now,

${}^n{C}_r{+}^n{C}_{r-1}$

$=\frac{n!}{r!(n-r)!}$$+\frac{n!}{(r-1)!(n-r+1)!}$ [ Using the definition of ${}^n{C}_r$]

$=\frac{n!}{(n-r)!(r-1)!}[\frac{1}{r}$$+\frac{1}{(n-r+1)}]$

$=\frac{n!}{(n-r)!(r-1)!}\left[\frac{(n+1)}{r(n+1-r)}\right]$

$=\frac{(n+1)!}{r!(n+1-r)!}$

${=}^{n+1}{C}_r$.