Product Rule for Counting/Examples/Choices from 4, 3 and 2

Example of Use of Product Rule for Counting
Let $N$ be the number of ways you can choose at least $1$ item of fruit from:
 * $4$ (indistinguishable) oranges
 * $3$ (indistinguishable) bananas
 * $2$ (indistinguishable) apples

Then:
 * $N = 59$

Proof
You can choose:
 * $0$, $1$, $2$, $3$ or $4$ oranges, giving you $5$ options
 * $0$, $1$, $2$ or $3$ bananas, giving you $4$ options
 * $0$, $1$, or $2$ apples, giving you $3$ options

Each of these options is independent of each other.

Hence the Product Rule for Counting can be applied.

Hence the total number of ways you can choose a selection of fruit from the above is $5 \times 4 \times 3 = 60$.

This includes the option of choosing no items of fruit at all.

We specifically want to exclude that option.

Hence:
 * $N = 5 \times 4 \times 3 - 1 = 59$