I have \(5\) different pullover shirts and \(4\) different button-down shirts. In how many ways can I choose shirts for the next \(9\) days if I insist on wearing pullover shirts two days in a row at least once? Assume that I wear one shirt each day, and every shirt gets worn once.
Given no restrictions except that a different shirt must be worn every day.
There are 9! different possibilities = 362880
I think that the only way there there are NOT going to be at least 2 pullover shirts worn in a row is if the pullover and the button down shirts are worn on alterate days.
There are 5!*4! = 2880 ways to do this.
So I think there are 9! - 5!4! = 362880 - 2880 = 360,000 ways to do this to wear the shirts given that 2 pullovers must be worn in a row at least once.