If 1 + x n = C 0 + C 1 x + C 2 x 2 + . . . . + C n x n and ∑ r = 0 50 C r 2 r + 1 = m ! n ! 2 , then the value of m + n is equal to (where C r represents C r n )

Step-by-step solution

1 + x 50 = C 0 + C 1 x + C 2 x 2 + . . . . . + C 50 x 50 = ∑ r = 0 50 C r x r ∫ 0 x 1 + x 50 d x = ∑ r = 0 50 ∫ 0 x C r x r d x 1 + x 51 - 1 51 = ∑ r = 0 50 C r x r + 1 r + 1 and x + 1 50 = ∑ r = 0 50 C r x 50 - r Multiplying these two equations, we get, 1 + x 51 - 1 51 x + 1 50 = ∑ r = 0 50 C r x r + 1 r + 1 ∑ r = 0 50 C r x 50 - r 1 + x 101 - 1 + x 50 51 = C 0 x 1 + C 1 x 2 2 + C 2 x 3 3 + . . . . + C 50 x 51 51 C 0 x 50 + C 1 x 49 + C 2 x 48 + . . . . + C 50 Now, c