Permutations & Combinations
Factorial and its Properties
Grade 11
Question:
<p>Find <i>n</i>, if <i>(n + 2)! = 60 × (n - 1)!</i></p>
Step-by-Step Solution
Key Concept: Expand factorial expressions using the property (n+2)! = (n+2)(n+1)n(n-1)! and simplify to get a cubic equation in n.
<p><strong>Step 1:</strong> We know that <i>(n + 2)! = (n + 2)(n + 1)n(n - 1)!</i></p><p><strong>Step 2:</strong> Therefore, $$\frac{(n + 2)!}{(n - 1)!} = (n + 2)(n + 1)n$$</p><p><strong>Step 3:</strong> Given that <i>(n + 2)! = 60 × (n - 1)!</i>, we have:<br/>$$60 = (n + 2)(n + 1)n$$</p><p><strong>Step 4:</strong> Testing values: <i>5 × 4 × 3 = (n + 2) × (n + 1) × n</i></p><p><strong>Step 5:</strong> Comparing, we get <i>n = 3</i></p><p>∴ <i>n = 3</i></p>
Correct Answer: 3