<p>If \((1+x)^n = C_0 + C_1 x + C_2 x^2 + \ldots + C_n x^n\), then the sum of the product of the coefficients taken two at a time can be represented by \(\displaystyle\sum_{i=0}^{n}\sum_{j=i+1}^{n} C_i C_j = 2^a - \dfrac{b!}{c(d!)^2}\). Then which of the following are correct?</p>
Step-by-Step Solution
Key Concept: The sum of products of binomial coefficients taken two at a time equals half of [(∑Cᵢ)² - ∑Cᵢ²]. Since ∑Cᵢ = 2ⁿ and ∑Cᵢ² = C(2n,n), we get the form 2^(2n-1) - C(2n,n)/2.
<p><strong>Step 1:</strong> Recognize that ∑ᵢ₌₀ⁿ ∑ⱼ₌ᵢ₊₁ⁿ CᵢCⱼ represents products of distinct pairs.</p><p><strong>Step 2:</strong> Use the identity: (∑Cᵢ)² = ∑Cᵢ² + 2∑ᵢ₌₀ⁿ ∑ⱼ₌ᵢ₊₁ⁿ CᵢCⱼ</p><p><strong>Step 3:</strong> Therefore: ∑ᵢ₌₀ⁿ ∑ⱼ₌ᵢ₊₁ⁿ CᵢCⱼ = [(∑Cᵢ)² - ∑Cᵢ²]/2</p><p><strong>Step 4:</strong> Substitute ∑Cᵢ = 2ⁿ (from x=1) and ∑Cᵢ² = C(2n,n) (Vandermonde's identity with r=n):</p><p>Result = [2²ⁿ - C(2n,n)]/2 = 2^(2n-1) - C(2n,n)/2 = 2^(2n-1) - (2n)!/[2·(n!)²]</p><p>∴ a = 2n-1, b = 2n, c = 2, d = n</p>
Correct Answer: A