Question:

<p><strong>Step 1:</strong> Rewrite the expression as (abc + abd + acd + bcd)^{10}. Each term is a product of three variables from {a,b,c,d}.</p><p><strong>Step 2:</strong> Denote the four terms as T₁ = abc, T₂ = abd, T₃ = acd, T₄ = bcd. We need: (T₁)^{x₁}(T₂)^{x₂}(T₃)^{x₃}(T₄)^{x₄} where x₁ + x₂ + x₃ + x₄ = 10.</p><p><strong>Step 3:</strong> Expand in terms of variables: (abc)^{x₁}(abd)^{x₂}(acd)^{x₃}(bcd)^{x₄} = a^{x₁+x₂+x₃} b^{x₁+x₂+x₄} c^{x₁+x₃+x₄} d^{x₂+x₃+x₄}.</p><p><strong>Step 4:</strong> Set up the system of equations for coefficient a^4 b^8 c^9 d^9:<br>• x₁ + x₂ + x₃ = 4 (power of a)<br>• x₁ + x₂ + x₄ = 8 (power of b)<br>• x₁ + x₃ + x₄ = 9 (power of c)<br>• x₂ + x₃ + x₄ = 9 (power of d)<br>• x₁ + x₂ + x₃ + x₄ = 10 (total selections)</p><p><strong>Step 5:</strong> From equation 4: x₂ + x₃ + x₄ = 9. From equation 5: x₁ = 10 - 9 = 1.</p><p><strong>Step 6:</strong> From equation 1: 1 + x₂ + x₃ = 4, so x₂ + x₃ = 3.</p><p><strong>Step 7:</strong> From equation 4: x₂ + x₃ + x₄ = 9, and x₂ + x₃ = 3, so x₄ = 6.</p><p><strong>Step 8:</strong> From equation 2: 1 + x₂ + 6 = 8, so x₂ = 1.</p><p><strong>Step 9:</strong> From x₂ + x₃ = 3 and x₂ = 1: x₃ = 2.</p><p><strong>Step 10:</strong> Verify: x₁ = 1, x₂ = 1, x₃ = 2, x₄ = 6. Check all equations:<br>• 1 + 1 + 2 = 4 ✓<br>• 1 + 1 + 6 = 8 ✓<br>• 1 + 2 + 6 = 9 ✓<br>• 1 + 2 + 6 = 9 ✓<br>• 1 + 1 + 2 + 6 = 10 ✓</p><p><strong>Step 11:</strong> The coefficient is the multinomial coefficient: C(10; 1,1,2,6) = 10!/(1!·1!·2!·6!) = (10·9·8·7·6!)/(1·1·2·6!) = (10·9·8·7)/2 = 2520.</p><p><strong>∴ Answer: C</strong></p>