Simplify: \(^{n}C_{r} ÷ ^{n}C_{r-1}\)

A.

\(\frac{n(n-r)}{r}\)

B.

\(\frac{n}{r(n-r)}\)

C.

\(\frac{1}{r(n-r)}\)

D.

\(\frac{n+1-r}{r}\)

Correct answer is D

\(^{n}C_{r} = \frac{n!}{(n-r)! r!}\)

\(^{n}C_{r - 1} = \frac{n!}{(n - (r - 1))! (r - 1)!}\)

\(^{n}C_{r} ÷ ^{n}C_{r - 1} = \frac{n!}{(n - r)! r!} ÷ \frac{n!}{(n-(r-1))!(r-1)!}\)

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

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

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

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