<p>If the number of terms in the expansion of <span>\((x + y + z)^n\)</span> are 36, then find the value of <span>\(n\)</span>.</p>
Step-by-Step Solution
Key Concept: The number of terms in the expansion of (x + y + z)^n equals the number of non-negative integer solutions to a + b + c = n, which is given by the formula C(n+2, 2) = (n+1)(n+2)/2.
<p><strong>Step 1:</strong> In the expansion of (x + y + z)^n, each term has the form x^a·y^b·z^c where a + b + c = n and a, b, c are non-negative integers.</p><p><strong>Step 2:</strong> The number of such terms equals the number of non-negative integer solutions to a + b + c = n, which is C(n+2, 2) = (n+2)!/(2!·n!) = (n+1)(n+2)/2.</p><p><strong>Step 3:</strong> Given that the number of terms is 36:</p><p>(n+1)(n+2)/2 = 36</p><p>(n+1)(n+2) = 72</p><p><strong>Step 4:</strong> Expanding: n² + 3n + 2 = 72</p><p>n² + 3n - 70 = 0</p><p><strong>Step 5:</strong> Factoring: (n + 10)(n - 7) = 0</p><p>Since n must be positive: n = 7</p><p><strong>Verification:</strong> (7+1)(7+2)/2 = 8×9/2 = 36 ✓</p><p>∴ Answer: n = 7</p>
Correct Answer: n = 7