Permutations & Combinations
Arrangements with restrictions
Grade 11
Question:
<p>If the best and the worst papers never appear together, find in how many ways six examination papers can be arranged.</p>
Step-by-Step Solution
Key Concept: Use complementary counting: subtract arrangements where best and worst papers appear together from total arrangements. Treat the pair of inseparable papers as a single unit.
<p><strong>Step 1:</strong> Find total arrangements of 6 papers without any restriction.</p><p>Total arrangements = 6! = 720</p><p><strong>Step 2:</strong> Find arrangements where best and worst papers appear together.</p><p>Treat best and worst papers as a single unit. Now we have 5 units to arrange.</p><p>Arrangements of 5 units = 5! = 120</p><p>Within the unit, best and worst can be arranged in 2! = 2 ways</p><p>∴ Arrangements where best and worst appear together = 5! × 2! = 120 × 2 = 240</p><p><strong>Step 3:</strong> Apply complementary counting.</p><p>Arrangements where best and worst papers never appear together = Total - (Best and worst together)</p><p>= 720 - 240 = 480</p><p>∴ Answer: 480</p>
Correct Answer: 480