Whenever you get stuck on a counting problem, you can attempt to solve it, by making a smaller scale example where you can actually list all of the possible answers.  Then after doing a listing, you will probably catch on to a general method for counting all of the possibilities for a problem of that type.  A couple of students have asked me questions about the last problem of quiz 15.

Let's try a smaller scale example of that.

Suppose you have a child that wants some desert, but you insist that they eat at 4 servings of fruit and vegetables first, with at least one of the servings being a vegetable.  You have 5 fruits to choose from and 3 vegetables to choose from.

Thus the child can choose to eat:

1. 1 vegetable and 3 fruits
2. 2 vegetables and 2 fruits
3. 3 vegetables and 1 fruit.

Now suppose the vegetable choices are: Asperagaus, Broccolli, and Carrots.  Suppose the fruit choices are: Grapes, Oranges, Pineapple, Raspberries, and Watermelon.

For each case above, let's list the possibilities by using the first letters of the fruits and vegetables and an underscore to separate the vegetables from the fruits

1.  A_GOP    A_GOR     A_GOW    A_GPR     A_GPW     A_GRW    A_OPR    A_OPW     A_ORW     A_PRW
    B_GOP    B_GOR     B_GOW     B_GPR     B_GPW     B_GRW    B_OPR     B_OPW     B_ORW     B_PRW
    C_GOP    C_GOR     C_GOW     C_GPR    C_GPW     C_GRW    C_OPR     C_OPW     C_ORW     C_PRW
2. Now you do the same thing for this case
3. and again for this case.

As you can see, I got 30 possibilities for the first case.  Can you tell how we would have calculated that without listing all of the possiblilites out?  Can you do the same for the second and third case?

When you are finished with all of the cases, you should end up with an answer of 65 total cases, which is a lot, but not too many to list by hand.