Answer
a. 720
b. 36
c. $\displaystyle \frac{1}{20}$
Work Step by Step
"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)=$\displaystyle \frac{36}{720}=\frac{1}{20}$
