99 Percentile Qs Bank for JEE MainMathematicsBinomial Theorem
If n , r are two positive integers such that 1 r < n , then ^n P_ r+1 +r^2 n-1 P_ r-1 +(r+1)^ n-1 P_r+r^ n-1 P_ r-1 =
Options
- A^ n+2 P_ r+2
- B^ n+2 P_ r+1
- C( n +1) !
- D^ n+1 P_ r+1
Correct answer
D. ^ n+1 P_ r+1
Step-by-step solution
^n P_ r+1 +r^2 ^ n-1 P_ r-1 +(r+1) ^ n-1 P_r+r ^ n-1 P_ r-1 aligned & = ^n P_ r+1 +(r+1) ^ n-1 P_r+r^2 ^ n-1 P_ r-1 +r ^ n-1 P_ r-1 & = n ! (n-r-1) ! +(r+1) (n-1) ! (n-r-1) ! +r^2 (n-1) ! (n-r) ! +r (n-1) ! (n-r) ! & = n !(n-r)+(r+1)(n-1) !(n-r)+r^2(n-1) !+r(n-1) ! (n-r) ! & = (n-1) ! (n(n-r)+(r+1)(n-r)+r^2+r ) (n-r) ! & = (n-1) ! (n^2-n r+n r-r^2+n-r+r^2+r ) (n-r) ! & = (n-1) ! (n^2+n ) (n-r) ! = n(n+1)(n-1) ! (n-r) ! & = (n+1) ! (n-r) ! = ^ n+1 P_ r+1 aligned