I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket.)

65

Step-by-step explanation:

First, let's consider the case where you have 1 apple in the basket. The possible number of baskets you can have with 1 apple is 11: the case where you have no oranges besides the apples, the case where you have 1 orange, the case where you have 2 orange, etc.

This line of reasoning can be followed by each possible amount of apples you can have: 0, 1, 2, 3, 4 and 5.

Keep in mind that in the case where you have no apples, you must exclude the case where you also have no oranges, because the problem tells you that the basket must have at least 1 fruit.

So, the answer would be:

x = 6*11 - 1 = 65 possible baskets