Problem: There are 10 people in a line, where each person is either male or female. How many different lineups are there, where there are either 5 consecutive men, or 6 consecutive women?
Here's my answer:
For the case of 5 consecutive men: say you have the lineup M M M M M _ _ _ _ _ then for the other 5 spaces to the right you can either choose a man or a woman. Using the product rule, there are 2^5 different ways to create a lineup with 5 consecutive men in the first 5 spots. Now if you shift the 5 consecutive men 6 times to the right, you get a total of 6 * 2^5 different ways to create a lineup with 5 consecutive men.
For the case of 5 consecutive women: This would be the same situation as above and we would be able to create a total of 6 * 2^5 different ways to create a lineup with 5 consecutive women.
But there are 2 cases where there are both 5 consecutive men and 5 consecutive women such as: M M M M M W W W W W and W W W W W M M M M M
So the answer for the amount of different lineups is 12*2^5 - 2
is my answer correct?