<p><strong>For Problems 12–14:</strong> Consider the expansion of \((a + b + c + d)^6\). Then the sum of all the coefficients of the terms</p><p><strong>14.</strong> Which contains both \(a\) and \(b\) is</p>
Step-by-Step Solution
Key Concept: To find coefficients of terms containing both a and b in (a+b+c+d)^6, use the complementary counting method: subtract cases where a or b is absent from the total expansion.
<p><strong>Step 1:</strong> Sum of all coefficients in (a+b+c+d)⁶ is found by setting a=b=c=d=1: (1+1+1+1)⁶ = 4⁶ = 4096</p><p><strong>Step 2:</strong> Use inclusion-exclusion. Let A = terms missing a, B = terms missing b.</p><p>Terms missing a: (b+c+d)⁶ with a=1 gives 3⁶ = 729</p><p>Terms missing b: (a+c+d)⁶ with b=1 gives 3⁶ = 729</p><p>Terms missing both a and b: (c+d)⁶ with a=b=1 gives 2⁶ = 64</p><p><strong>Step 3:</strong> By inclusion-exclusion, terms containing <strong>both</strong> a and b:</p><p>Sum = 4⁶ - 3⁶ - 3⁶ + 2⁶ = 4096 - 729 - 729 + 64 = 2702</p><p>Note: If the answer key states 4, the question likely asks for a different quantity (such as the number of distinct terms or a specific coefficient). With the problem as stated, the sum of coefficients containing both a and b is <strong>2702</strong>. Please verify the exact wording, as answer 4 may refer to the exponent sum or a counting problem rather than the coefficient sum.</p>
Correct Answer: 4