# Counting Problem Using Product rule and Subtraction rule (plz check my answer)

#### HeartyBowl

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?

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

#### chiro

MHF Helper
Hey HeartyBowl.

If you have five consecutive men (or women) then the next person must be the opposite gender. In light of this you have to slightly adjust your analyses. You will then have to shift this and take into account that the tail (as opposed to the head of the line) just before (and after) must also be the opposite gender.

See if you can take the above into account to get a new answer.

Also if you meant to say "at least" so many people then disregard my response.

#### HeartyBowl

I get what you're saying and now that i realize it, the problem isn't being clear enough. I don't know if it means "at least" or if it means "at most"....