Question

Prove that 1 + 1. ¹P₁ + 2. ²P2 + 3. ³P₃ + …. n. ⁿP_n = ⁿ⁺¹P_n+1.

Answer 1

To Prove 1 + 1. ¹P₁ + 2. ²P2 + 3. ³P₃ + …. n. ⁿP_n = ⁿ⁺¹P_n+1.

Formula Used: 

Total number of ways in which n objects can be arranged in r places (Such that no object is replaced) is given by

 ⁿP_r = \(\frac{n!}{(n-r)!}\)

1 + 1. ¹P₁ + 2. ²P₂ + 3. ³P₃ + …. n. ⁿP_n = ⁿ⁺¹P_n+1. 

1 + (2! - 1!) + (3! - 2!) + (4! - 3!) + …….((n + 1)! - n!) = (n + 1)! 

1 + ((n + 1)! - 1!) = (n + 1)! 

(n + 1)! = (n + 1)! 

Hence proved.