<p>If the coefficient of 4th term in the expansion of <span>\((a + b)^n\)</span> is 56, then find the value of <span>\(n\)</span>.</p>
Step-by-Step Solution
Key Concept: The coefficient of the rth term in (a+b)^n is C(n,r-1). Here the 4th term has coefficient C(n,3) = 56, so we solve nC3 = 56 to find n.
<p><strong>Step 1:</strong> Identify the coefficient of the 4th term in (a+b)^n.</p><p>The general term is T_{r+1} = C(n,r)a^{n-r}b^r</p><p>The 4th term is T_4 = T_{3+1}, so r = 3</p><p>Coefficient of 4th term = C(n,3) = 56</p><p><strong>Step 2:</strong> Use the binomial coefficient formula.</p><p>C(n,3) = n!/(3!(n-3)!) = n(n-1)(n-2)/(3×2×1) = n(n-1)(n-2)/6 = 56</p><p><strong>Step 3:</strong> Solve for n.</p><p>n(n-1)(n-2) = 336</p><p>Testing values: When n = 8: 8×7×6 = 336 ✓</p><p><strong>Step 4:</strong> Verify the answer.</p><p>C(8,3) = 8×7×6/6 = 56 ✓</p><p>∴ Answer: n = 8</p>
Correct Answer: n = 8