Autor: 06.12.2023
Combinatorics - The Multiplication Rule
We will be exploring the multiplication rule, but first, let's clarify what combinations are.
Combinations involve selecting a specific number of elements from a set where the order doesn't matter. They determine how we create sets from different things, regardless of their sequence. It's nothing more than choosing a certain number of elements from a set where order is not significant.
The Multiplication Rule
The multiplication rule in combinatorics is a principle that allows us to calculate the number of possible combinations of two or more choices. We need to multiply the number of possible outcomes for each of these choices.
If we have two independent decisions to make, where the first has 3 possible outcomes and the second has 4 possible outcomes, the overall number of possible combinations for these two decisions is 12 (3 * 4).
Real-life example: choosing clothes
Let's say you have 3 types of shirts (A, B, C) and 2 types of pants (X, Y). You want to figure out how many different outfits you can put together by choosing one shirt and one pair of pants.
- Shirts: A, B, C (3 possible choices),
- Pants: X, Y (2 possible choices).
According to the multiplication rule, the number of different outfits is 3 (number of shirts) * 2 (number of pants) = 6 different outfits.
Verification - are we calculating everything correctly?
Let's verify step by step whether the number of combinations is indeed 6. We'll manually list all the possible combinations of our clothing.
Here are the outfits:
- Shirt A + Pants X (1)
- Shirt A + Pants Y (2)
- Shirt B + Pants X (3)
- Shirt B + Pants Y (4)
- Shirt C + Pants X (5)
- Shirt C + Pants Y (6)
It seems we've calculated everything correctly; we have 6 possible outfits.
Summary
The multiplication rule is very useful in combinatorics for calculating the number of possible outcomes when there are several independent decisions or actions to be made.
