<p>The sum of the last eight coefficients in the expansion of \((1 + x)^{16}\) is equal to</p>
Step-by-Step Solution
Key Concept: The binomial coefficients are symmetric: C(16,r) = C(16,16-r). The last 8 coefficients are C(16,9) through C(16,16), which by symmetry equal C(16,0) through C(16,7). Use the fact that the sum of all coefficients is 2^16, and by symmetry, each half sums to 2^15.
<p><strong>Step 1:</strong> Identify the last 8 coefficients in (1+x)^16.</p><p>The expansion has 17 terms with coefficients C(16,0), C(16,1), ..., C(16,16).</p><p>The last 8 coefficients are: C(16,9), C(16,10), C(16,11), C(16,12), C(16,13), C(16,14), C(16,15), C(16,16).</p><p><strong>Step 2:</strong> Apply symmetry property of binomial coefficients.</p><p>By symmetry: C(16,k) = C(16,16-k)</p><p>Therefore: C(16,9)=C(16,7), C(16,10)=C(16,6), ..., C(16,16)=C(16,0)</p><p>So the sum of last 8 coefficients = C(16,0) + C(16,1) + ... + C(16,7)</p><p><strong>Step 3:</strong> Use the sum of all coefficients.</p><p>Sum of all coefficients = 2^16 (put x=1 in (1+x)^16)</p><p>By symmetry, sum of first 8 = sum of last 8 = 2^16/2 = 2^15</p><p><strong>Step 4:</strong> Calculate.</p><p>2^15 = 32,768</p><p>∴ Answer: A (2^15 or 32,768)</p>
Correct Answer: A