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.

$7$ Choices from $4$

Product Rule for Counting/Examples/7 Choices from 4