Exams › JEE Advanced › Maths

What is the coefficient of x⁵⁰ in the expansion of the sum S = sum(r=0 to 100) [ C(100,r) * (x-3)^(100-r) * 2^r ]?

1. C(100,50) * 2⁵⁰
2. C(100,51) * 2⁵¹
3. C(100,50)
4. C(100,51) * 2⁵⁰

Correct answer: C(100,50)

Solution

The sum equals (x-1)¹⁰⁰ by the binomial theorem. The coefficient of x⁵⁰ in (x-1)¹⁰⁰ is C(100,50)*(-1)⁵⁰ = C(100,50).

Related JEE Advanced Maths questions

• In the expansion of (1 + x)ⁿ, let the binomial coefficients be represented as C₀, C₁, C₂,..., Cₙ. If p and q are such that p + q = 1, what is the value of Σᵣ₌₀ nCr pʳqⁿ⁻ʳ?
• What is the result of the summation Σ₀≤k≤n ∑ᵢ i * Cᵢ?
• The expression E5(n) is defined as Σ (nCk × k⁴) / Σ (nCk × k³), and it simplifies to n / 4. For n ≤ 109, E5(n) is less than 26. Therefore, the smallest value of n that satisfies the condition E5(n) ≥ 26 is one of the following:
• What is the coefficient of x¹¹ in the expansion of (1 + x²)⁴ (1 + x³)⁷ (1 + x)¹²?
• Evaluate the sum T = sum(r=0 to 20) r*(20-r)*[C(20,r)]².
• What is the coefficient of x² in the expansion of (1 + x)⁵ * (1 + 2x)⁴?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →