The coefficient of x¹⁵ in the expansion of _ r=0 ²⁰ x^r (1+x)^ 40-r is

Step-by-step solution

The given expression is a geometric progression with 21 terms: S = (1+x)⁴⁰ + x(1+x)³⁹ + x^2(1+x)³⁸ + + x²⁰(1+x)²⁰ Here, the first term is a = (1+x)⁴⁰ and the common ratio is r = x 1+x . Using the sum formula for a GP, S = a 1 - r^n 1 - r : S = (1+x)⁴⁰ 1 - ( x 1+x )²¹ 1 - x 1+x Simplifying the denominator: 1 - x 1+x = 1 1+x Substituting this back: S = (1+x)⁴⁰ (1+x)²¹ - x²¹ (1+x)²¹ 1 1+x S = (1+x)⁴⁰ (1+x)²¹ - x²¹ (1+x)²⁰ S = (1+x)²⁰ [ (1+x)²¹ - x²¹ ] S = (1+x)⁴¹ - x²¹(1+x)²⁰ We need to find the coefficient of x¹⁵ in