Permutations & Combinations
Pairing and Grouping
Grade 11
Question:
<p>The total number of ways in which \(2n\) persons can be divided into \(n\) couples, is</p>
<p>(a) \(\frac{(2n)!}{(n!)^2}\)</p>
<p>(b) \(\frac{(2n)!}{(2n!)^n}\)</p>
<p>(c) \(\frac{(2n)!}{n!(2n!)^2}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The number of ways to partition $2n$ distinct objects into $n$ indistinct pairs is $\frac{(2n)!}{2^n \cdot n!}$.
<p>Dividing $2n$ persons into $n$ indistinguishable couples: we partition into pairs, which is $\frac{(2n)!}{2^n \cdot n!}$. (Divide by $2^n$ for the indistinguishability within pairs and by $n!$ for the indistinguishability of couples.)</p>
Correct Answer: D