A certain candy store sells jellybeans in the following six flavors only: banana, chocolate, grape, lemon, peach and strawberry. The jellybeans are sorted into boxes containing exactly 2, 3 or 4 different flavors, with each possible assortment of flavors appearing in exactly one box. What is the probability that any given box contains grape jellybeans?

As I see, possible assortments are:

(2,2,2), (4,2), (3,3).

I think I have to find the number of ways to form each of those assortments, then divide that by the number of boxes that contain grapes?

This sound a bit overly complicated to me - could someone point me in the right direction?

• You mean there is one box for each 2-combination of flavours, one for each 3-combo and one for each 4-combo? Commented Jun 30, 2019 at 10:58

First you should count the total number of boxes using the binomial coefficient. For example: How many possibilities are there to sort 6 balls in boxes of two? Answer: $$\binom{6}{2}$$. Do that for all 3 types of boxes und you have the total ammount of boxes. Then you count the number of boxes the grape jellybeans are in. For example: In the boxes of two there are 5 boxes with grape jellybeans in it. You get to that conclusion by fixing one spot to the grape jellybean and counting, how many possibilities there are to fill the remaining spots. Count all possibilities and you have got your answer by dividing the number of boxes with grape jellybeans by the total number of boxes.

• > "You get to that conclusion by fixing one spot to the grape jellybean and counting, how many possibilities there are to fill the remaining spots". I don't understand this part...so we have 15 boxes, each containing two flavors. We want to find the number of these boxes that contain grape jeallybean. In one of these boxes, we fix one spot for grape, so there are 5 spots remaining for the other spot...I sort of see it, but it isn't intuitively clear to me Commented Jun 30, 2019 at 11:09
• Maybe this will clarify it: You want to count the number of boxes with 3 flavours that contain the grape jellybean. So you know that one of the 3 spots is taken by the grape flavor. You have 2 remaining spots and 5 flavors to fill them. How many possibilities are there to do so?
– Emu
Commented Jun 30, 2019 at 11:14
• so there would be 5*4 possibilities....ah I think I see it now after drawing it out :) thanks! Commented Jun 30, 2019 at 11:17