I'm having some trouble with a math problem.

I have 28 different flavors of ice cream and I can only have 3 scoops. I can repeat and the order is partially important. There are 9 fruit flavors.

The question is:

What is the possible number of combinations of any three-scoop ice cream cone if the first flavor had to be a fruit flavor?

I'm not entirely sure what the formula for this would even be.


The flavor of the first scoop can be selected in $9$ ways.

I will interpret the order is partially important to mean that we do not care in which order the second and third scoops are selected.

If the second and third scoops are the same flavor, we have $28$ choices.

If the second and third scoops are different flavors, we have $\binom{28}{2}$ choices since we are selecting a subset of two of the $28$ flavors.

$$9\left[28 + \binom{28}{2}\right]$$

If the order did matter, we would have $9$ choices for the first scoop and $28$ choices for both the second and third scoops.

$$9 \cdot 28^2$$


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