Question:

<p><strong>Step 1:</strong> Simplify the base fraction using geometric series formula.</p><p>$$\frac{1-t^6}{1-t} = 1 + t + t^2 + t^3 + t^4 + t^5 = \sum_{k=0}^{5} t^k$$</p><p><strong>Step 2:</strong> Cube this expression.</p><p>$$\left(\frac{1-t^6}{1-t}\right)^3 = (1 + t + t^2 + t^3 + t^4 + t^5)^3$$</p><p><strong>Step 3:</strong> Find coefficient of t⁴ in the triple product. We need all ways to pick three terms from {1, t, t², t³, t⁴, t⁵} whose exponents sum to 4.</p><p><strong>Step 4:</strong> The partitions of 4 into three non-negative integers ≤ 5 are:</p><p>• (0,0,4): coefficient = $\binom{3}{1} = 3$</p><p>• (0,1,3): coefficient = $\binom{3!}{1!1!1!} = 6$</p><p>• (0,2,2): coefficient = $\binom{3}{1,2} = 3$</p><p>• (1,1,2): coefficient = $\binom{3!}{2!1!} = 3$</p><p><strong>Step 5:</strong> Total coefficient = 3 + 6 + 3 + 3 = <strong>15</strong></p>