<p>The coefficient of \(x^9\) in the expansion of \((1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})\) is _______.</p>
Step-by-Step Solution
Key Concept: The coefficient of x⁹ equals the number of ways to write 9 as a sum of distinct positive integers ≤ 100, where selecting (1+xᵏ) contributes xᵏ to the product. We need to count partitions of 9 into distinct parts.
<p><strong>Step 1:</strong> Expanding $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$ means choosing either 1 or $x^k$ from each factor $(1+x^k)$.</p><p><strong>Step 2:</strong> The coefficient of $x^9$ counts ways to select a subset $S \subseteq \{1,2,3,\ldots,100\}$ such that $\sum_{k \in S} k = 9$.</p><p><strong>Step 3:</strong> Find all partitions of 9 into distinct positive integers $\leq 100$:</p><ul><li>$9 = 9$ ✓</li><li>$9 = 1 + 8$ ✓</li><li>$9 = 2 + 7$ ✓</li><li>$9 = 3 + 6$ ✓</li><li>$9 = 4 + 5$ ✓</li><li>$9 = 1 + 2 + 6$ ✓</li><li>$9 = 1 + 3 + 5$ ✓</li><li>$9 = 2 + 3 + 4$ ✓</li></ul><p><strong>Step 4:</strong> Verify no other combinations work. Any partition with 4+ parts requires minimum sum $1+2+3+4=10 > 9$.</p><p>∴ Answer: <strong>8</strong></p>
Correct Answer: 8