Combinations with repetition calculator

An ice cream parlour has eight flavours and you want three scoops. You could choose three different flavours, two chocolate and one pistachio, or three vanilla. The order does not matter: a cone is the same ice cream whichever scoop is on top. How many options are there? That is exactly what combinations with repetition count: the number of ways to choose k items from n types when types may repeat.

If the flavours could not be repeated, these would be ordinary combinations, and the binomial coefficient calculator would do the job.

How it works

The calculator uses the formula

$$\left((\genfrac{}{}{0ex}{}{n}{k})\right)=(\genfrac{}{}{0ex}{}{n+k-1}{k})$$

The double brackets on the left are read “n multichoose k”.

Picture the three scoops from eight flavours as a row of three circles and seven dividers that separate the flavours. The circles before the first divider are vanilla, those between the first and second dividers are chocolate, and so on. Each choice of ice cream corresponds to exactly one arrangement, so we just need to count the ways of choosing three of the ten positions for the circles:

$$\left((\genfrac{}{}{0ex}{}{8}{3})\right)=(\genfrac{}{}{0ex}{}{8+3-1}{3})=(\genfrac{}{}{0ex}{}{10}{3})=\frac{10\cdot 9\cdot 8}{3\cdot 2\cdot 1}=120$$

There are 120 ways to make three scoops from eight flavours. A detailed derivation, using the example of buying bottles of beer for a camping trip, is in the article on combinations with repetition.