Ice cream combinatorics question

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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.]

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2 Answers

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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.

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$3=3$ giving $5$ possibilities. (all $3$ have the same flavour)

$3=2+1$ giving $5\times4=20$ possibilities. ($2$ with the same flavour and $1$ with another)

$3=1+1+1$ giving $\binom{5}{3}=10$ possibilities. (all $3$ have different flavour)

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