An ice cream shop sells ice creams in $5$ possible flavours: vanilla, chocolate, strawberry, mango and pineapple.
How many combinations of $3$ scoops cone are possible? [note: repetition of flavours is allowed, but the order in which they are chosen does not matter.]
Best Answer
This can be solved using the strategy used in this post:
It comes down to calculating $$ \binom{5+3-1}{3}=\binom 73=\frac{7\cdot6\cdot5}{1\cdot2\cdot3}=35 $$
What we are doing is the following: We create $5$ 'buckets' of flavors: $$ ---|---|---|---|--- $$
Now, we want do divide the three scoops over the $5$ buckets. Below, a scoop is shown as $O$. Some possibilities are: $$ O|O|||O\\ ||OOO||\\ etc... $$ There are $4$ borders between the buckets and $3$ scoops. Thus, there are $\binom{4+3}{3}=35$ possibilities.