Question

Find $r$ if ${}^{15}{C}_{3r}={}^{15}{C}_{r+3}$

Answer 1

${}^n{\text{C}}_r{=}^n{\text{C}}_{n-r}$${}^n{\text{C}}_r{+}^n{\text{C}}_{r-1}{=}^{(n+1)}{\text{C}}_r$ If ${}^n{\text{C}}_r{=}^n{\text{C}}_s$then $r=s$ or $r+s=n$

Given, ${}^{15}{C}_{3r}={}^{15}{C}_{r+3}$
${\Rightarrow }^{15}{C}_{15-3r}={}^{15}{C}_{r+3}$
$\Rightarrow 15-3r=r+3\Rightarrow 3r=12\Rightarrow r=3$