## Task B-2.5.

Ana is visiting her grandmother and wants to bring her a few pieces of fruit in a basket. She has 6 bananas, 5 apples, and 4 peaches available.

a) In how many ways can she choose the fruit for the basket if the basket must not be empty?

b) In how many ways can she choose the fruit for the basket if she must include at least one piece of each type of fruit?

(Note: the arrangement of the fruit is not important, only which fruits are in the basket and how many of each.)
## Solution.

a) Ana can choose:

- zero, one, two, three, four, five, or six bananas, which is 7 possibilities,
- zero, one, two, three, four, or five apples, which is 6 possibilities,
- zero, one, two, three, or four peaches, which is 5 possibilities.

Thus, there are $7 \cdot 6 \cdot 5=210$ possibilities for choosing the number of pieces of fruit.

Since we have included the possibility of having 0 pieces of each type of fruit in our counting, we need to subtract this possibility, so the desired number of ways is 209.

b) If the basket must contain one piece of fruit of each type, the number of possibilities is $6 \cdot 5 \cdot 4=120$.