Product Rule for Counting/Examples/Choices from 4, 3 and 2
Jump to navigation
Jump to search
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:
- $N = 5 \times 4 \times 3 - 1 = 59$
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $8$