<p>(a) If \({}^{22}P_{r+1} : {}^{20}P_{r+2} = 11:52\), find \(r\).</p>
Step-by-Step Solution
Key Concept: Expand permutations using the formula P(n,r) = n!/(n-r)! and simplify the ratio by canceling factorials, then solve the resulting equation systematically.
<p><strong>Step 1:</strong> Write the permutation formula for the given ratio.</p><p>$${}^{22}P_{r+1} : {}^{20}P_{r+2} = 11:52$$</p><p>$$\frac{{}^{22}P_{r+1}}{{}^{20}P_{r+2}} = \frac{11}{52}$$</p><p><strong>Step 2:</strong> Expand using P(n,r) = n!/(n-r)!.</p><p>$$\frac{\frac{22!}{(22-r-1)!}}{\frac{20!}{(20-r-2)!}} = \frac{11}{52}$$</p><p>$$\frac{22!}{(21-r)!} \times \frac{(18-r)!}{20!} = \frac{11}{52}$$</p><p><strong>Step 3:</strong> Simplify 22!/20! = 22 × 21 = 462.</p><p>$$\frac{462 \times (18-r)!}{(21-r)!} = \frac{11}{52}$$</p><p><strong>Step 4:</strong> Expand (21-r)! = (21-r)(20-r)(19-r)(18-r)!.</p><p>$$\frac{462}{(21-r)(20-r)(19-r)} = \frac{11}{52}$$</p><p><strong>Step 5:</strong> Cross multiply and simplify.</p><p>$$462 \times 52 = 11(21-r)(20-r)(19-r)$$</p><p>$$24024 = 11(21-r)(20-r)(19-r)$$</p><p>$$(21-r)(20-r)(19-r) = 2184$$</p><p><strong>Step 6:</strong> Test values. If r = 7: (14)(13)(12) = 2184 ✓</p><p>∴ Answer: r = 7</p>
Correct Answer: 7