Permutations & Combinations
Selecting and Arranging
Grade 11
Question:
<p>There are <i>n</i> married couples at a party. Each person shakes hand with every other person other than her or his spouse. The total number of handshakes must be</p>
<p>(A) \(\binom{2n}{2} - 2n\)</p>
<p>(B) \(\binom{2n}{2} - n\)</p>
<p>(C) 2<i>n</i>(2<i>n</i> – 1)</p>
<p>(D) 2<i>n</i>(<i>n</i> – 1)</p>
Step-by-Step Solution
Key Concept: Start with all possible handshakes and subtract those between spouses.
<p><strong>Analysis:</strong> Total people = 2<i>n</i>. Without restriction, total handshakes = \(\binom{2n}{2} = \frac{2n(2n-1)}{2} = n(2n-1)\)</p><p>Married couples do not shake hands with each other. There are <i>n</i> such pairs to exclude.</p><p>Total handshakes = \(n(2n-1) - n = n(2n - 1 - 1) = n(2n - 2) = 2n(n-1)\)</p>
Correct Answer: D