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: Option D
Explanation
(^{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})