<p>The value of \({}^{15}C_0^2 - {}^{15}C_1^2 + {}^{15}C_2^2 - \cdots - {}^{15}C_{15}^2\) is</p>
Step-by-Step Solution
Key Concept: Use the identity that ∑(−1)^r (nCr)² = (−1)^(n/2) · nC(n/2) when n is even, derived from the coefficient of x^n in (1−x²)^n · (1+x)^n = (1−x)^n · (1+x)^n.
<p><strong>Step 1:</strong> Recognize the pattern as an alternating sum of squared binomial coefficients: ∑(−1)^r (15Cr)² for r = 0 to 15.</p><p><strong>Step 2:</strong> Use the generating function identity. The coefficient of x^15 in (1−x)^15(1+x)^15 = (1−x²)^15 equals ∑(−1)^r (15Cr)².</p><p><strong>Step 3:</strong> Expand (1−x²)^15. The term x^15 cannot appear since all powers are even. Alternatively, use the direct formula: for n = 15 (odd), ∑(−1)^r (nCr)² = 0.</p><p><strong>Step 4:</strong> Since 15 is odd, the coefficient of x^15 in (1−x²)^15 is 0 (no odd powers exist).</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: C