<p>If the difference of the number of arrangements of three things from a certain number of dissimilar things and the number of selections of the same number of things from them exceeds 100, then the least number of dissimilar things is</p>
Step-by-Step Solution
Key Concept: Set up the inequality P(n,3) - C(n,3) > 100 and simplify using factorial expressions to find the minimum integer value of n that satisfies it.
<p><strong>Step 1:</strong> Write the given condition as an inequality.</p><p>P(n,3) - C(n,3) > 100</p><p><strong>Step 2:</strong> Express using factorial formulas.</p><p>P(n,3) = n(n-1)(n-2)</p><p>C(n,3) = n(n-1)(n-2)/6</p><p><strong>Step 3:</strong> Substitute into the inequality.</p><p>n(n-1)(n-2) - n(n-1)(n-2)/6 > 100</p><p><strong>Step 4:</strong> Factor out common terms.</p><p>n(n-1)(n-2)[1 - 1/6] > 100</p><p>n(n-1)(n-2) × (5/6) > 100</p><p>n(n-1)(n-2) > 120</p><p><strong>Step 5:</strong> Test integer values.</p><p>For n = 5: 5(4)(3) = 60 ✗</p><p>For n = 6: 6(5)(4) = 120 ✗ (not exceeding)</p><p>For n = 7: 7(6)(5) = 210 > 120 ✓</p><p><strong>Step 6:</strong> Verify the original condition for n = 7.</p><p>P(7,3) - C(7,3) = 210 - 35 = 175 > 100 ✓</p><p>∴ The least number of dissimilar things is <strong>7</strong></p>
Correct Answer: C