Permutations & Combinations
Grade 11
Question:
<p>If <sup>n+2</sup>C<sub>8</sub> : <sup>n-2</sup>P<sub>4</sub> = <span class="math-tex">\(\frac{57}{16}\)</span>, then n is equal to</p>
<p style="display:inline">21</p>
<p style="display:inline">5</p>
<p style="display:inline">20</p>
<p style="display:inline">19</p>
Step-by-Step Solution
Key Concept: Simplify the ratio using factorial definitions of combinations and permutations to reduce the equation to a product of consecutive integers.
<p><sup>n+2</sup>C<sub>8</sub> : <sup>n-2</sup>P<sub>4</sub> = <span class="math-tex">$\frac{57}{16}$</span><br />
<span class="math-tex">$\Leftrightarrow \frac{(n+2) !}{8 !(n-6) !} \times \frac{(n-6) !}{(n-2) !}=\frac{57}{16}$</span><br />
<span class="math-tex">$\Leftrightarrow \frac{(n+2)(n+1) n(n-1)(n-2) !}{8 !(n-2) !}=\frac{57}{16}$</span><br />
<span class="math-tex">$\Leftrightarrow$</span> (n + 2) (n + 1) n (n - 1)<br />
<span class="math-tex">$=\frac{57}{16}$</span> <span class="math-tex">$\times$</span> 8 <span class="math-tex">$\times$</span> 7 <span class="math-tex">$\times$</span> 6 <span class="math-tex">$\times$</span> 5 <span class="math-tex">$\times$</span> 4 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 2 <span class="math-tex">$\times$</span> 1<br />
= 19 <span class="math-tex">$\times$</span> 3 <span class="math-tex">$\times$</span> 7 <span class="math-tex">$\times$</span> 6 <span class="math-tex">$\times$</span> 5 <span class="math-tex">$\times$</span> 4 <span class="math-tex">$\times$</span> 3<br />
= 19 <span class="math-tex">$\times$</span> 21 <span class="math-tex">$\times$</span> 18 <span class="math-tex">$\times$</span> 20<br />
<span class="math-tex">$\Leftrightarrow$</span> (n + 2) (n + 1) n (n - 1) = 21 <span class="math-tex">$\times$</span> 20 <span class="math-tex">$\times$</span> 19 <span class="math-tex">$\times$</span> 18<br />
<span class="math-tex">$\Rightarrow$</span> n = 19</p>
Correct Answer: D