First slide

Combinations

Question

The number of ways of dividing 15 men and 15 women into 15 couples, each consisting of a man and a woman, is

Moderate

A

1240
B

1840
C

1820
D

2005

Solution

The number of ways of choosing first couple is $(^{15} C_{1}) (^{15} C_{1}) = 15^{2}$ . The number of ways of choosing 2nd couple is $(^{14} C_{1}) (^{14} C_{1}) = 14^{2}$ , and so on. Thus, the number of ways of choosing the couples is
$\begin{array}{l} 15^{2} + 14^{2} + 13^{2} + \dots + 2^{2} + 1^{2} \\ = \frac{15 \times (15 + 1) [2 (15) + 1]}{6} = 1240 \end{array}$

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