Suppose you have 5 blueberry pancakes, 2 banana pancakes, and 2 chocolate chip pancakes. How many ways can you stack the pancakes by flavour?

Question

Suppose you have 5 blueberry pancakes, 2 banana pancakes, and 2 chocolate chip pancakes. How many ways can you stack the pancakes by flavour?

Answer 1

Your full stack will be 9 pancakes high. So where will the blueberry pancakes be? You have $9\choose 5$ possibilities to choose their position (because you cannot distinguish two blueberry pancakes from one another). So now you have only 4 places left. You have now $4\choose 2$ possibilities to choose positions for the two banana pancakes and the two chocolate chip pancakes will have to take the last 2 remaining places in your stack. So the total amount of ways to stack the pancakes by flavour is $n={9\choose 5}*{4\choose 2} = 756$