How many different flavour combinations are possible at Peters’?

Peters’ Drive In is famous in Alberta for their milkshakes. They have three base flavours to choose from. However, they advertise that as many as three flavours can be combined in a milkshake, resulting in an advertised >4,500 combinations. The only question is: are they lying?

In order to confirm the advertised flavour possibilities, I was asked to ‘Write a list of the flavours and count them’. Really, it’s a simple task. The list must be comprised of flavour possibilities using 1, 2, or 3 base flavours, it cannot use duplicate flavours (eg. strawberry, coconut, strawberry), and the order of the flavours does not matter (we are counting combinations, not permutations).

I wrote a VBA macro for my minion computer, instructing it to make a complete menu. It spit out a list of 4,525 flavours. The list was then printed on an 8.5×11″ sheet of paper (double-sided, no line spacing, font size 2) and will be hand delivered to the person who asked for a written list.

It should be noted that this list accounts for all combinations which include the Banana base flavour. Having removed such a terrible flavour from the base flavour list, the total number of combinations decreased to 4,089. It’s terrible to imagine that Peters’ will can provide almost 500 Banana flavour combinations – Yuck!