Question

If ²⁰C_r = ²⁰C_{r + 4} , then ^rC₃ is equal to
(a) 54
(b) 56
(c) 58
(d) none of these

Answer 1

(b) 56
$r+r+4=20$ [∵${}^n{C}_x{=}^n{C}_y\Rightarrow n=x+y\mathrm{or}x=y$]
$$\begin{gathered} \Rightarrow 2r+4=20 \\ \Rightarrow 2r=16 \\ \Rightarrow r=8 \end{gathered}$$

Now,${}^r{C}_3={}^8{C}_3$
${}^8{C}_3=\frac{8!}{3!5!}=\frac{8\times 7\times 6}{3\times 2\times 1}=56$