Probability with seating

[chelsea88]

Four married couples have reserved eight seats in a row at the theater, starting at an isle seat. If they arrange themselves randomly, what is the probability that all the women will sit in adjacent seats and all the men will sit in adjacent seats?

I know the denominator is 8! but I am not sure what the probability is for each seat. I also know the first seat would have 8 choices, but I am not sure about the second seat. Would it be 3 possibilities because all of one gender has to sit next to eachother? I need help figuring out the possibilities for the seats.

 

[soroban]

Hello, chelsea88!

Four married couples have reserved eight seats in a row at the theater.
If they arrange themselves randomly, what is the probability that
all the women will sit in adjacent seats and all the men will sit in adjacent seats?

\(\displaystyle \text{There are }\,8!\,=\,40,\!320\text{ possible seating arrangements.}\)

\(\displaystyle \text{Arranging by gender, there are }\,\boxed{2}\, \text{ possible cases: }\;MMMMWWWW\,\text{ or }\,WWWWMMMM\)

\(\displaystyle \text{In each case, the 4 men can be arranged in }\boxed{4!}\text{ orders}\)
. . \(\displaystyle \text{and the 4 women can be arranged in }\boxed{4!}\text{ orders.}\)

\(\displaystyle \text{Hence, there are: }\:2\times 4! \times 4! \:=\:1152\,\text{ ways that the genders are separated.}\)


\(\displaystyle \text{Therefore: }\;P(\text{men together, women together}) \;=\;\frac{1152}{40,\!320} \;=\;\frac{1}{35}\)

 

[chelsea88]

Okay, that makes sense, that was a very good explanation for me. Thank you!