1 An ice cream shop has 31 different flavors of ice cream an

1.

An ice cream shop has 31 different flavors of ice cream and two kinds of cones. The rules are, cones can be one, two or three-scoops (but no larger), and if two scoops are the same kind of ice cream they are indistinguishable. Otherwise the order of the scoops matters. How many different ice cream cones can a customer order at this shop?

Solution

An ice cream shop has 31 different flavors of ice cream and two kinds of cones. The rules are, cones can be one, two or three-scoops (but no larger), and if two scoops are the same kind of ice cream they are indistinguishable. Otherwise the order of the scoops matters. How many different ice cream cones can a customer order at this shop?

If they order 1 scoop ice cream cone, with 31 different ice creams and 2 types of cones, there are 31*2 = 62 difference ice creams.

If they order a 2 scoop ice cream cone:

If they get 2 different flavors, there are 31*30*2 = 1860 cones.

If they get the same flavors, there are 31*2 = 62 cones.

If they order a 3 scoop ice cream cone:

If they get 3 different flavors, there are 31*30*29*2 = 53940 cones.

If they get 2 different flavors, there are (31C2)*6*2 = 5580 cones (explanation: 31 c 2 - pick 2 flavors out of 31, 6 ways to arrange 2 favors, for example, VVC, VCV, CVV, CCV, CVC, VCC, and 2 choices of cones)

If they get the same flavor for all three scoops, there are 31*2 = 62 cones.

Add them up: 62 + 1860 + 62 + 53940 + 5580 + 62 = 61566 different cones