The total number of terms in the expansion of (x+y)⁶⁰+(x-y)⁶⁰ is

Step-by-step solution

The expansion of (x+y)^ n is given by _ k=0 ^ n ^ n C_ k x^ n-k y^ k . The expansion of (x+y)⁶⁰ is ⁶⁰C₀ x⁶⁰ + ⁶⁰C₁ x⁵⁹ y + ⁶⁰C₂ x⁵⁸ y^2 + + ⁶⁰C₆₀ y⁶⁰ . The expansion of (x-y)⁶⁰ is ⁶⁰C₀ x⁶⁰ - ⁶⁰C₁ x⁵⁹ y + ⁶⁰C₂ x⁵⁸ y^2 - + ⁶⁰C₆₀ y⁶⁰ . Adding these two expressions, the terms with odd powers of y cancel out: (x+y)⁶⁰ + (x-y)⁶⁰ = 2 [ ⁶⁰C₀ x⁶⁰ + ⁶⁰C₂ x⁵⁸ y^2 + ⁶⁰C₄ x⁵⁶ y^4 + + ⁶⁰C₆₀ y⁶⁰ ] . The terms remaining correspond to k = 0, 2, 4, , 60 . The number of such terms is 60 - 0 2 + 1 = 30 + 1 = 31 . Answer: 31