# Product Rule for Counting/Examples

## 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.