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In the expansion of the product P(x) = (x - 1)(x² - 2)(x³ - 3)... (xⁿ - n) where n >= 30, the coefficient of x^((n² + n - 14)/2) is:

1. 13
2. -13
3. 7
4. -7

Correct answer: 13

Solution

Each factor (x^k - k) contributes either x^k (power k) or the constant (-k). To get total power n(n+1)/2 - 7, we must choose the constant term from a subset S of factors where the sum of indices = 7. The coefficient is the sum over all such subsets of the product of (-k). Subsets summing to 7: {7}->(-7)=-7; {1,6}->(-1)(-6)=6; {2,5}->(-2)(-5)=10; {3,4}->(-3)(-4)=12; {1,2,4}->(-1)(-2)(-4)=-8. Total = -7+6+10+12-8 = 13.

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