In a bowl of ice cream, the order of the scoops does not matter. a) suppose you can make 630 two-scoop bowls of ice cream, each containing two different flavours, at the shop where you work. how many flavours of ice cream are available in this shop? b) how many two-scoop bowls could you make if you can duplicate flavours?

Part aanswer: 36  let n be the number of flavours.the question says the order does not matter, so we use combinations.we choose 2 flavours from a set of n flavours and we're supposed to have 630 total combinations. so the equation to use here is    use definition of combinations to deal with the left side, i.e.    doing this, we have    because of how factorial works, we can expand n! into n(n-1)! or n(n-1)(n-2)! or n(n-1)(n-2)(n-3)! ; we can expand it out as much as we want. let's go up to (n-2)! because that will result in cancellation with the denominator.    can solve by quadratic formula:     reject -35 because negative flavours make no sense.so we have 36 flavours in this shop.part banswer: 1296  from part a, we have 36 flavours to choose from. so by the fundamental counting principle, choosing two flavours with duplicate flavours allowed is    36  · 36 = 1296