Thinking Mathematically (6th Edition)

a. 720 b. 36 c. $\displaystyle \frac{1}{20}$
"Lining up" implies importance of order, that is, permutations. In (1,2,3,4,5,6), we replace each number with a person's name to obtain a permutation of all six people in the line. a. 6 people in line can make ${}_{6}P_{6}=6!=720$ permutations. b. Positions 1,3,5 must be filled with women. This can be done in ${}_{3}P_{3}=3!=6$ ways Positions 2,4,6 must be filled with men. This can be done in ${}_{3}P_{3}=3!=6$ ways. The total number of ways this can be done is $3!\cdot 3!=36$ c. Let E be the described event. P(E)$=\displaystyle \frac{the\ number\ of\ ways\ the\ permutation\ can\ occur}{total\ number\ of\ possible\ permutations}$ P(E)=$\displaystyle \frac{36}{720}=\frac{1}{20}$