Permutations & Combinations
Permutations
Grade 11
Question:
<p>Find the value of <i>r</i>, if <span style="math">\(^{22}P_{r+1} : \,^{20}P_{r+2} = 11 : 52\)</span></p>
Step-by-Step Solution
Key Concept: Expand permutation notation using the formula and equate coefficients to find the unknown variable.
<p><strong>Solution:</strong></p><p>Given: <span style="math">$\frac{^{22}P_{r+1}}{^{20}P_{r+2}} = \frac{11}{52}$</span></p><p>Expanding: <span style="math">$\frac{22 \times 21 \times \,^{20}P_{r-1}}{(19-r)(20-r)(21-r) \cdot \,^{20}P_{r-1}} = \frac{11}{52}$</span></p><p>Simplifying: <span style="math">$\frac{22 \times 21}{(19-r)(20-r)(21-r)} = \frac{11}{52}$</span></p><p><span style="math">$(21-r)(20-r)(19-r) = 52 \times 2 \times 21 = 14 \times 13 \times 12$</span></p><p>Therefore, <span style="math">$r = 7$</span></p>
Correct Answer: 7