Different ways to order 3 scoops of ice cream and order them depending on cone or bowl and other conditions.

Question

I was just wondering if someone can let me know if my solutions are correct and/or if I am approaching the problems the correct way.

a)If the 3 scoops of ice cream are being placed in a bowl.... I decided to approach this with multichoosing so I had 3 multichoose 31 which is the equivalent of 31 stars and 2 bars.

Therefore I had 33C31 which I evaluated to 528

b)now for the 3 scoops of ice cream are being placed in a cone where the order of the ice cream in the cone matters I was thinking 31x31x31x3. Since for each of the scoops there are 31 flavors or options and then times 3 to evaluate the different ordering of the 3 scoops chosen.

However I've come to realize can't I order 3 scoops of different ice cream 6 ways?Also what if two flavors are repeated then ordering becomes more technical. Should this question also be approached with multi choosing. Help would be greatly appreciated.

Answer 1

How many ways are there to have $3$ scoops of ice cream placed in a bowl?

Your answer is correct.

How many ways are there to have $3$ scoops of ice cream placed on a cone?

Each possible cone can be viewed as a choice of bottom, middle, and top scoops. There are $31$ choices for each level of cone, so there are $31^3$ possible ways to order a $3$-scoop cone.