Permutations & Combinations
Properties of Combinations
Grade 11
Question:
<p>If <span class="math">\(\binom{n}{9} = \binom{n}{7}\)</span>, find <span class="math">\(n\)</span>.</p>
Step-by-Step Solution
Key Concept: When two binomial coefficients with the same n but different r values are equal, the sum of the r values equals n.
<p><strong>Step 1:</strong> Use the property that if <span class="math">$\binom{n}{r} = \binom{n}{s}$</span> and <span class="math">r \neq s$</span>, then <span class="math">r + s = n$</span>.</p><p><strong>Step 2:</strong> We have <span class="math">$\binom{n}{9} = \binom{n}{7}$</span> where <span class="math">9 \neq 7$</span>.</p><p><strong>Step 3:</strong> Therefore, <span class="math">$n = 9 + 7 = 16$</span>.</p><p>∴ <span class="math">n = 16</span></p>
Correct Answer: 16