Permutations And Combinations Question 157

Question: If $ ^{n}C _{r} $ denotes the number of combinations of n things taken r at a time, then the expression $ ^{n}{C _{r+1}}+{{}^{n}}{C _{r-1}}+2\times {{}^{n}}C _{r} $ equals [AIEEE 2003]

Options:

A) $ ^{n+2}C _{r} $

B) $ ^{n+2}{C _{r+1}} $

C) $ ^{n+1}C _{r} $

D) $ ^{n+1}{C _{r+1}} $

Show Answer

Answer:

Correct Answer: B

Solution:

  • Expression = $ ^{n}{C _{r+1}}+{{}^{n}}{C _{r-1}}+{{}^{n}}C _{r}+{{}^{n}}C _{r} $ = $ ^{n}{C _{r+1}}+{{}^{n+1}}C _{r}+{{}^{n}}C _{r} $ = $ ^{n+1}{C _{r+1}}+{{}^{n+1}}C _{r} $ = $ ^{n+2}{C _{r+1}} $ .