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JEE MainMathematicsBinomial Theorem

Let C be the coefficient of x³⁰ in the expansion of _ k=0 ³⁰ x^k (1+x)^ 50-k . If C = m n ⁵⁰C₃₀ , where m and n are co-prime positive integers, then the value of m + n is

Correct answer

24

Step-by-step solution

The given series is a geometric progression: S = (1+x)⁵⁰ + x(1+x)⁴⁹ + x^2(1+x)⁴⁸ + + x³⁰(1+x)²⁰ The first term is a = (1+x)⁵⁰ , the common ratio is r = x 1+x , and the number of terms is 31 . The sum of the GP is given by S = a 1 - r³¹ 1 - r S = (1+x)⁵⁰ 1 - ( x 1+x )³¹ 1 - x 1+x S = (1+x)⁵⁰ (1+x)³¹ - x³¹ (1+x)³¹ 1 1+x S = (1+x)²⁰ ( (1+x)³¹ - x³¹ ) S = (1+x)⁵¹ - x³¹(1+x)²⁰ We need to find the coefficient of x³⁰ in S . In the first term, (1+x)⁵¹ , the coefficient of x³⁰ is ⁵¹C₃₀ . In the second term, x³¹(1+x)²⁰ , the

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