Multiplication method

Multiplication method (also called the product rule) is one of the most fundamental counting techniques.

It is used when we have to combine several choices at the same time.

 

The method is based on the idea that if we have several independent choices, we can find the total number of combinations by multiplying the number of possibilities for each choice.

 

 

Example 1: Clothing choice

A student has to put together an outfit:

 

• 3 shirts
• 2 pairs of pants
• 4 pairs of shoes

 

Since you have to choose shirt and pants and shoes, we must multiply the possibilities:

 

$$ \large 3 \cdot 2 \cdot 4 = 24\ combinations $$

 

 

Example 2: Password

A password consists of 3 digits, where each digit can be from 0 to 9:

 

• 1st digit: 10 possibilities
• 2nd digit: 10 possibilities
• 3rd digit: 10 possibilities

 

In total we get:

 

$$ \large 10 \cdot 10 \cdot 10 = 1000\ codes $$

 

 

Example 3: Two dice

We roll two standard dice. Each die has 6 possible outcomes:

 

• First die: 6 possibilities
• Second die: 6 possibilities

 

In total there are:

 

$$ \large 6 \cdot 6 = 36\ outcomes $$

 

 

Formula

In general:

 

$$ \large \text{Number of combinations} = a \cdot b \cdot c \cdot \ldots \cdot n $$

 

 

When can you use the Multiplication method?

 

• When you have to make several choices at the same time.
• When the individual choices are independent of each other.

 

A simple tree diagram can illustrate the method. For each choice, the possibilities branch out, and the total number of branches corresponds to the product of the possibilities.

 

 

When the choices are not independent

The Multiplication method only works if the choices are independent. If they influence each other, we cannot simply multiply the possibilities.

 

Example: If you have to create a code with 3 digits, but the digits cannot be repeated, the number of possibilities is:

 

$$ \large 10 \cdot 9 \cdot 8 = 720 $$

 

 

Summary

The Multiplication method is used when you have to choose several things at the same time. The total number of combinations is found by multiplying the number of possibilities for each choice.

 

The method is therefore a foundation in combinatorics and plays a central role in topics such as permutations, combinations and probability theory.