4 flavors of ice cream, 15 candy toppings, 4 liquid syrups. How many different combinations are possible?

Question

1-4 scoops of icream (1 of each flavor), 0-15 toppings, 0-4 syrups.

Answer 1

I’ve made certain assumptions:

Ice cream is mandatory, but topping and syrup are optional;

I can have between 1 and 4 scoops of ice cream, but no more than 1 of each flavour;

There can only be at most one topping and there are 15 different flavours;

There can only be at most one syrup and there are 4 different flavours;

The choice of ice cream, topping and syrup are independent.


ICE CREAM

1 scoop: 4 because there are 4 flavours

2 scoops: 6

3 scoops: 4

4 scoops: 1 because no more than one of each flavour is permitted

TOTAL FOR ICE CREAM: 15

TOPPING: 16 including no topping

SYRUP: 5 including no syrup.

Total combination: 15*16*5=1200.

(If the initial assumption were changed to allow up to 15 toppings instead of just one, the numbers involved would be too large to handle. The situation is worse if more than one syrup is allowed. So the initial assumption seems to be the practical, sensible one!)