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We had a few solutions sent in including one that got to the heart of the matter and produced a really good account. This is it from Jacob in the United Kingdom .

We started with the maximum number of combinations of ice cream that you can have with four flavours of ice cream. We counted $15$ combinations:

The first person would have all four flavours.

4 more people could then have three of the flavours.

6 more people could have two flavours.

4 more people could have one flavour.

$1 + 4 + 6 + 4 = 15$

We added in another flavour of ice cream, which gave us five flavours. We worked out that there were $31$ combinations for five flavours:

The first person would have all five flavours.

Five more people could then have four of the flavours.

Ten more people could have three flavours.

Ten more people could have two flavours.

Five more people could have one flavour.

$1 + 5 + 10 + 10 + 5 = 31$

We noticed a pattern that we had seen before! This pattern is in a number triangle where you add up the two numbers in the row above. When we added up the total of each row in the number triangle, they added up to one more than the number of ice cream combinations!

We then used the number triangle to work out how many ice cream combinations there would be for six flavours, seven flavours, eight flavours, nine flavours and ten flavours.

When we had worked out these combinations, we also realised that each total in the number triangle was the previous total multiplied by two. This made it very quick to work out how many ice cream combinations there would be for any number of flavours of ice cream!

My Dad helped me with building the number table on Excel and attaching it to this solution.

I really enjoyed finding the patterns in this problem.

Thank you for sending in your solution to us, Jacob. It looks like you worked hard on this investigation.