The I Scream Shop specializes in triple scoop sundaes. How many different triple scoop sundaes could be made using these flavors: vanilla, chocolate, strawberry, cherry, lime, and lemon?
I attempted to solve this problem by breaking it down into three cases: The first case involves the same flavor 3 times in a row. The second is 2 of a kind and then another flavor. The third is for ice cream containing three different flavors.
EDIT: The order of the scoops does not matter for each ice cream served.
The main problem I have is that I feel my work is not only incorrect, but also for the cases that I did manage to find an answer, I wish to find a more concrete answer instead of counting.
Case $1$: For three different ice creams, we have $\binom63= 20$
Case $2$: This case we can choose two different flavors. There are $\binom62$ ways of doing this. Now we must consider which one of these flavors doubles. At this point, I had a gut feeling that it is twice as much, but I'm not to sure how to do this for $n$ scoops.
Anyways, the total here is $\binom62\times{2}= 30$
Case $3$: We choose one flavor, so there is $\binom61= 6$ ways.
Therefore, there is a total of $20+30+6=56$ possible ice cream scoops. Again, my main concern is if there is a more concrete and general way to do the second case, or in the general, the cases between the two extremes- choosing one and choosing all different.