# Product Rule for Counting/Examples

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## Examples of Use of Product Rule for Counting

### Choices from $2$ and $3$

The canonical example concerns choices from the menu at a restaurant:

You may select exactly one dish from each category:

- Starters

- $(1): \quad$ Crottled Greeps
- $(2): \quad$ Stone Soup
- $(3): \quad$ Petty-Dwarf Roots

- Main Course

- $(1): \quad$ Hufu Salad
- $(2): \quad$ Braised Trake in Funistrada

The diner then has $2 \times 3 = 6$ possible different meals:

- $(1): \quad$ Crottled Greeps with Hufu Salad
- $(2): \quad$ Crottled Greeps with Braised Trake in Funistrada

- $(3): \quad$ Stone Soup with Hufu Salad
- $(4): \quad$ Stone Soup with Braised Trake in Funistrada

- $(5): \quad$ Petty-Dwarf Roots with Hufu Salad
- $(6): \quad$ Petty-Dwarf Roots with Braised Trake in Funistrada

### Choices from $4$, $3$ and $2$

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:

- $N = 59$

### Choices from $5$ and $3$

There are:

- $5$ different ways to travel from $A$ to $B$
- $3$ different ways to travel from $B$ to $C$.

Hence there are $5 \times 3 = 15$ different ways to travel from $A$ to $C$.

### $6$ Football Matches

Let it be understood that a football match between two teams $\text A$ and $\text B$ can end as:

- A win for team $\text A$
- A draw
- A loss for team $\text A$.

That being understood, then there are $729$ ways to predict the results of $6$ football matches.