Permutations & Combinations
Grade 11
Question:
<p><span class="math-tex">\(\frac{100}{10!}=\frac{1}{8!}+\frac{x}{9!}\)</span>. Find x .</p>
<p style="display:inline">1</p>
<p style="display:inline">2</p>
<p style="display:inline">3</p>
<p style="display:inline">4</p>
Step-by-Step Solution
Key Concept: Simplify the equation by expressing all factorials in terms of the smallest factorial present using the property n! = n(n-1)! to allow for cancellation.
<p><span class="math-tex">$\frac{100}{10!}=\frac{1}{8!}+\frac{x}{9!}$</span>.<br />
We know, <span class="math-tex">$n!=n \cdot(n-1) .(n-2) .(n-3) \ldots \ldots . . . . . . . .=n(n-1)!$</span><br />
<span class="math-tex">$\frac{100}{10.9 .8!}=\frac{1}{8!}+\frac{x}{9.8!}$</span><br />
<span class="math-tex">$\Rightarrow \frac{100}{10.9}=\frac{1}{1}+\frac{x}{9}$</span><br />
<span class="math-tex">$\Rightarrow>\frac{10}{9}=1+\frac{x}{9}$</span><br />
<span class="math-tex">$\Rightarrow>\frac{x}{9}=\frac{10}{9}-1=\frac{1}{9}$</span><br />
<span class="math-tex">$\Rightarrow x=1$</span>.</p>
Correct Answer: A