If you’ve read the combination and permutation posts, have a go at the following questions. These combinations/permutations questions are about as hard as any combinations or permutations questions you could see test day. Don’t forget to use the dash method!
1. Sally owns five different blouses. If she cannot wear the same blouse on two consecutive days, how many different ways can she dress on Wednesday and Thursday?
2. Quentin has five textbooks on his shelf. If chemistry must always be in the middle, how many ways can he arrange the books?
3. Tyler wants to choose an outfit for dinner. If he can choose from 3 shirts, 5 pairs of shoes, and 2 dress pants, how many different outfits can Tyler wear?
1. The best method is to use the dashes. The first dash can stand for Wednesday and the second for Thursday. For Wednesday Sally can wear any of the five blouses. On Thursday she can only choose from four different blouses. Remember she can’t wear blouses on consecutive days. This gives us: 5 x 4 = 20. Answer (D).
2. The trick here is to imagine that the chemistry book is invisible. After all, it is stuck in one place so it can’t be switched around with the other books. So it’s as though we only have four books. How many ways can we arrange four books? 4! = 24 Answer (A).
3. Tyler is only choosing 1 of each article of clothing. While we could use the combination formula as the first step, whenever you are choosing for only one thing, the number of possibilities is the number you are choosing from. So if he can choose one shirt from 3 shirts, the total number of shirts he can select is 3. Using the dash method, we can place a number in each dash: 3 x 5 x 2 = 30. Answer (C).