total ways to arrange 9*8*7= 504

why not 9c3= 84 ways

Basic principle of counting method

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I also used to struggle with when to just write out ___ X ___ X ___ and when to use the formula. In this case, consider what the formula means "9 choose 3" But we are not really choosing 3. We are choosing 1 per room, or better said, we are choosing any of them for a room and then any of the left ones and then a final time. Perhaps someone else can give an example of this same question reworded so that the formula would be needed? Perhaps if we had 9 fossils, one display room and this one room was going to display 3 fossils at the same time?

If the question was to just select 3, then one could use 9C3 but since this can be arranged differently in the three different rooms, it is 9P3 = 504

Another way to think about this is to first come up with the number of unique groups of dinosaurs that will be selected. This is 9C3. But for each unique group of dinosaurs, they can be arranged in different rooms to create different or unique arrangements. Each group of 3 dinosaurs selected can be arranged 3! ways. Therefore, the total amount of ways is actually 9C3 * 6! = 504.