154360

Let's do this in general. Suppose you want as many kids as possible to eat some flavour(s) from the ice-cream cafe and there are x flavours. What is the maximum number of kids that they can all eat ice-cream, that nobody has exactly the same combination of other kids? Let's first find out the pattern. Let's say there is one flavour: A. Directly, you can see there are 2 possibilities: Eat A, Do not eat A. But not eating A means eating nothing in this case. So, only Child 1 can eat ice-cream: Flavour A. Now, Let's say there are 2 flavours: A and B. Then, Child 1 could eat A and B, Child 2 eats A, and Child 3 eats B. The next child, Child 4 can't eat anything. So, there are 3 possible ways for 2 flavours. As we saw, there are 7 ways from 3 flavours. And although it might take some time, there are 15 ways for 4 flavours. 1, 3, 7, 15.... Let's compare this with this sequence, starting from 2, doubling every time (these numbers in the sequence are known as powers of 2): 2, 4, 8, 16.... You can see that the terms in the first sequence are less than one than the second. The numbers 1, 3, 7, 15... has a special name; Mersenne numbers. And the way to generate them is to multiply 2 by itself x times, then subtract 1. This Ice Cream problem is related with the power set. A power set is all possible distinct ways that you can have some object(s). This includes the empty set, which has no objects. And the power set of x objects is 2 multiplied by itself x times, thus, 2 to the xth power.
But since we are ignoring the empty set, there are 1 less than 2 to the xth power ways for x flavours, which are the Mersenne numbers.