Let A = _ r=1 ⁵⁰ (2r-1) , ¹⁰⁰C_ 2r-1 . Let B be the maximum binomial coefficient in the expansion of (1+x)¹⁰¹ . If A ¹⁰⁰C₅₀ B = m n 2^l , where m and n are coprime odd positive integers, then the value of l + m + n is equal to:

Step-by-step solution

The sum A = _ r=1 ⁵⁰ (2r-1) , ¹⁰⁰C_ 2r-1 represents the sum of the odd coefficients multiplied by their indices. Consider the expansion (1+x)^n = _ k=0 ^n , ^ n C_ k x^k . Differentiating with respect to x , we get n(1+x)^ n-1 = _ k=1 ^n k , ^ n C_ k x^ k-1 . Substituting x=1 , we have n 2^ n-1 = _ k=1 ^n k , ^ n C_ k . Substituting x=-1 , we have 0 = _ k=1 ^n k , ^ n C_ k (-1)^ k-1 . Adding these two equations gives n 2^ n-1 = 2 _ k odd k , ^ n C_ k , which implies _ k odd k , ^ n C_ k = n 2^ n-2 . For n=100 , A =