Question:

<p><strong>Step 1:</strong> Simplify the base expression.</p><p>$$\frac{1-t^6}{1-t} = 1 + t + t^2 + t^3 + t^4 + t^5$$</p><p>This is a geometric series with first term 1, common ratio t, and 6 terms.</p><p><strong>Step 2:</strong> Find the coefficient of t⁴ in $(1 + t + t^2 + t^3 + t^4 + t^5)^3$.</p><p>We need all ways to select three terms (with repetition) from {1, t, t², t³, t⁴, t⁵} whose exponents sum to 4.</p><p><strong>Step 3:</strong> List all partitions of 4 into three non-negative integers ≤ 5:</p><p>• (0,0,4): coefficient = $\frac{3!}{2!1!} = 3$</p><p>• (0,1,3): coefficient = $\frac{3!}{1!1!1!} = 6$</p><p>• (0,2,2): coefficient = $\frac{3!}{1!2!} = 3$</p><p>• (1,1,2): coefficient = $\frac{3!}{2!1!} = 3$</p><p><strong>Step 4:</strong> Sum all coefficients.</p><p>$$3 + 6 + 3 + 3 = 15$$</p><p>∴ Answer: <strong>15</strong></p>