<p>The value of \(\displaystyle\sum_{r=2}^{10} {}^rC_2 \cdot {}^{10}C_r\) is</p>
Step-by-Step Solution
Key Concept: Rewrite ¹⁰Cᵣ as ¹⁰C₍₁₀₋ᵣ₎ and use the identity ʳC₂ = r(r-1)/2 to convert the sum into a form where you can extract coefficients from a binomial expansion. The trick is recognizing that ʳC₂ · ¹⁰Cᵣ counts specific paths in a combinatorial identity.
<p><strong>Step 1:</strong> Use the identity for ʳC₂:</p><p>ʳC₂ = r(r-1)/2, so the sum becomes:</p><p>∑(r=2 to 10) [r(r-1)/2] · ¹⁰Cᵣ = (1/2)∑(r=2 to 10) r(r-1) · ¹⁰Cᵣ</p><p><strong>Step 2:</strong> Apply the multiplication rule ¹⁰Cᵣ · r(r-1) = 10·9·⁸C₍ᵣ₋₂₎:</p><p>Since r · ¹⁰Cᵣ = 10 · ⁹C₍ᵣ₋₁₎ and (r-1) · ⁹C₍ᵣ₋₁₎ = 9 · ⁸C₍ᵣ₋₂₎</p><p>We get: r(r-1) · ¹⁰Cᵣ = 90 · ⁸C₍ᵣ₋₂₎</p><p><strong>Step 3:</strong> Substitute k = r-2, so r goes from 2 to 10 means k goes from 0 to 8:</p><p>(1/2) · 90 · ∑(k=0 to 8) ⁸Cₖ = 45 · 2⁸ = 45 · 256 = 11,520</p><p>∴ Answer: C</p>
Correct Answer: C