## Wednesday, June 27, 2012

### Counting Problem: Beginning Level

#1: Problem: An ice cream shop offers 5 special toppings. A customer may choose to add any of the following: caramel, gummy bears, marshmallow, pineapple and berries. How many different combinations are possible?

Since you can add any of the 5 things, that's make them A,B,C,D,E.

Solution I:
For not putting anything on ice cream. You have one choice.
For putting any one of the five toppings, you have 5 choices, which are A, B, C, D or E.
For any two, 10 choices, which are AB, AC, AD, AE, BC, BD, BE, CD, CE, DE.
For any three, again 10 choices.
For any four, 5 choices.
For putting all, 1 choice.
Sum them up and you get 32, which is the answer.

Have you noticed that it's the Pascal's triangle, 1 , 5 ,10 , 10 , 5 , 1
The same as 5C0 + 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 32

Solution II:
Another easier way to look at this question is for every choice, you can either put it on the sandwich or not putting it on, each thing 2 choices, so the answer is 25 = 32