Binomial Theorem
Sum involving binomial coefficients
Grade 11
Question:
<p>The value of \(\displaystyle\sum_{r=0}^{10} r\,{}^{10}C_r\,3^r(-2)^{10-r}\) is</p>
<p>(1) 20</p>
<p>(2) 10</p>
<p>(3) 300</p>
<p>(4) 30</p>
Step-by-Step Solution
Key Concept: Recognize that r·ⁿCᵣ = n·ⁿ⁻¹Cᵣ₋₁, then use the binomial theorem on (3-2)¹⁰ after index substitution to evaluate the sum directly.
<p><strong>Step 1:</strong> Use the identity r·ⁿCᵣ = n·ⁿ⁻¹Cᵣ₋₁</p><p>r·¹⁰Cᵣ = 10·⁹Cᵣ₋₁</p><p><strong>Step 2:</strong> Substitute into the sum:</p><p>∑ᵣ₌₀¹⁰ r·¹⁰Cᵣ·3ʳ·(-2)¹⁰⁻ʳ = 10∑ᵣ₌₀¹⁰ ⁹Cᵣ₋₁·3ʳ·(-2)¹⁰⁻ʳ</p><p><strong>Step 3:</strong> The r = 0 term vanishes. Reindex with s = r - 1:</p><p>= 10∑ₛ₌₀⁹ ⁹Cₛ·3ˢ⁺¹·(-2)⁹⁻ˢ</p><p><strong>Step 4:</strong> Factor out 3:</p><p>= 30∑ₛ₌₀⁹ ⁹Cₛ·3ˢ·(-2)⁹⁻ˢ</p><p><strong>Step 5:</strong> Apply binomial theorem: ∑ₛ₌₀⁹ ⁹Cₛ·3ˢ·(-2)⁹⁻ˢ = (3-2)⁹ = 1</p><p><strong>Step 6:</strong> Therefore: 30 × 1 = 30</p><p>∴ Answer: <strong>30</strong></p>
Correct Answer: D