1)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.
2)How many different non-congruent isosceles triangles can be formed by connecting three of the dots in a 4x4 square array of dots like the one shown below?
There are 9! ways with no restrictions and If 2 pullover shirts are NOT worn two days in a row then it will be
P B P B P B P B P
So the pullovers will have to be in the 1st 3rd 5th 7th and 9th positions these can be arranged in 5! ways. The button shirts can be arranged 4! ways
so that is5!*4! ways that two buttons up will not be together. So the number of ways that at least 2 pullovers are worn on consecutive days is
9! - 5!*4!
9!-5!*4! = 360000
Two triangles are congruent if they have the same traced outline, possibly up to rotating and flipping. This is equivalent to having the same set of 3 side lengths. SO the total is 9.
Confusedperson, if you use someone else's answer you need to give that person credit. If you find the answer somewhere else, please put a link to that source.