Question

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following : 

n.^{n – 1}C_{r – 1} = (n – r + 1)(ⁿC_{r – 1})

Answer 1

Given that we need to prove : 

n.^{n – 1}C_{r – 1} = (n – r + 1).ⁿC_{r – 1} 

Consider L.H.S, 

We know that,

ⁿC_r = \(\frac{n!}{r!(n-r)!}\)

And also,

n! = n(n – 1)(n – 2)…………2.1

= R.H.S 

∴ L.H.S = R.H.S, 

Thus proved.