Division rule
Division rule is one of the fundamental counting techniques. It is used when we have counted too many combinations, because the same solution can appear several times in the counting.
The rule is based on the fact that if each unique combination is counted the same number of times, we can find the correct number by dividing by this number.
Example 1: Seats
6 people must sit around a round table. If we count permutations directly, we get:
$$ \large 6! = 720 $$
But around a table, the arrangements are the same, even if everyone moves one seat to the right. Each unique arrangement is therefore counted 6 times. The correct number is:
$$ \large \frac{6!}{6} = 120 $$
Example 2: Colors in a flag
We want to make a flag with three stripes, using the colors red, white and blue. If we count all permutations, we get:
$$ \large 3! = 6 $$
But if two of the stripes have the same color, we get repetitions. For example: red, red, blue. Here each unique combination is counted several times. We correct this by dividing by the number of repetitions.
Formula
In general:
$$ \large \text{Number of unique combinations} = \frac{\text{number of counted combinations}}{\text{number of repetitions per combination}} $$
Especially for permutations with repetitions:
$$ \large \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} $$
where \( n \) is the total number of elements, and \( n_1, n_2, \ldots, n_k \) are the numbers of identical elements.
When can you use the Division rule?
- When the same combination appears several times in a counting.
- When arrangements or choices are considered the same in certain situations (symmetry or repetitions).
The Division rule is therefore a tool to correct an overcounting.
Summary
The Division rule is used when we count too many combinations because the same solution is counted several times. We find the correct number by dividing by the number of repetitions.
The rule plays an important role in permutations with repetitions and in situations with symmetry.