The total number of terms in the expansion of (x+y)¹⁰⁰+(x-y)¹⁰⁰ is

Step-by-step solution

The binomial expansion of (x+y)^ n is given by _ k=0 ^ n ^ n C_ k x^ n-k y^ k . Expanding the two terms: (x+y)¹⁰⁰ = ¹⁰⁰C₀ x¹⁰⁰ + ¹⁰⁰C₁ x⁹⁹ y + ¹⁰⁰C₂ x⁹⁸ y² + + ¹⁰⁰C₁₀₀ y¹⁰⁰ (x-y)¹⁰⁰ = ¹⁰⁰C₀ x¹⁰⁰ - ¹⁰⁰C₁ x⁹⁹ y + ¹⁰⁰C₂ x⁹⁸ y² - + ¹⁰⁰C₁₀₀ y¹⁰⁰ Adding these two expressions, the odd-indexed terms cancel out: (x+y)¹⁰⁰ + (x-y)¹⁰⁰ = 2 [ ¹⁰⁰C₀ x¹⁰⁰ + ¹⁰⁰C₂ x⁹⁸ y² + ¹⁰⁰C₄ x⁹⁶ y⁴ + + ¹⁰⁰C₁₀₀ y¹⁰⁰ ] The terms in the resulting expression correspond to k = 0, 2, 4, , 100 . The number of such terms is 100 - 0 2 + 1 = 50 + 1 = 51