Question

If ^mC₁ = ⁿC₂ , then
(a) 2 m = n
(b) 2 m = n (n + 1)
(c) 2 m = n (n − 1)
(d) 2 n = m (m − 1)

Answer 1

(c) 2 m = n (n − 1)

^mC₁ = ⁿC₂
$$\begin{gathered} \Rightarrow \frac{m!}{1!\left(m-1\right)!}=\frac{n!}{2!\left(n-2\right)!} \\ \Rightarrow \frac{m\left(m-1\right)!}{\left(m-1\right)!}=\frac{n\left(n-1\right)\left(n-2\right)!}{2\left(n-2\right)!} \\ \Rightarrow 2m=n\left(n-1\right) \end{gathered}$$