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Chapter 1: Problem 15

A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?

Short Answer

Expert verified
There are 24,024,640 possible results for choosing and pairing off the students in this dance class scenario.

Step by step solution

01

Selecting Men and Women

First, we need to determine the number of ways to select 5 men from 12 men, and 5 women from 10 women. We can use combinations for this purpose, as order does not matter when we are selecting the students. Number of ways to select 5 men out of 12 men is given by the combination: \(\binom{12}{5}\), and the number of ways to select 5 women out of 10 women is given by the combination: \(\binom{10}{5}\). Next, we will calculate the values of these combinations: For selecting men: \(\binom{12}{5} = \frac{12!}{5! (12-5)!} = \frac{12!}{5!7!} = 792\) For selecting women: \(\binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} = 252\)
02

Pairing the Chosen Men and Women

Now that we have chosen 5 men and 5 women, we need to pair them off. We can use permutations for this purpose, as order matters when we are pairing the students. There are 5 women and we need to pair them with 5 men. The number of ways to do this is given by the permutation: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
03

Calculating the Total Possible Results

Now we have the number of ways to select the men and women, and the number of ways to pair them off. To find the total possible results, we multiply these values together: Total possible results = (Ways to select men) 脳 (Ways to select women) 脳 (Ways to pair them) Total possible results = \(792 \times 252 \times 120 = 24,024,640\) So there are 24,024,640 possible results for choosing and pairing off the students in this dance class scenario.

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