Let K be the sum of the coefficients of the odd powers of x in the expansion of 1 + x 99 . Let a be the middle term in the expansion of 2 + 1 2 200 . If C 99 200 K a = 2 l m n , where m and n are odd numbers, then the ordered pair l , n is equal to:

Step-by-step solution

In the binomial expansion of 1 + x 99 , the sum of odd coefficients of x is given by K = C 1 + C 3 + … . + C 99 = 2 99 - 0 2 = 2 98 Also, a = Middle term in the binomial expansion of 2 + 1 2 200 is T 200 2 + 1 = T 100 + 1 = C 100 200 2 100 1 2 100 ⇒ T 100 + 1 = C 100 200 · 2 50 So, C 99 200 K a = C 99 200 × 2 98 C 100 200 × 2 50 = 100 101 × 2 48 = 25 101 × 2 50 = m n 2 l On comparing, we get l = 50 ,   m = 25   and   n = 101 Therefore, l ,   n ≡ 50 , 10