Ordered and unordered samples (combinations)
Sometimes one also considers whether the combinations that can be formed are ordered or unordered. If the combinations are ordered, the order matters. Conversely, the order does not matter if the combinations are unordered.

This means that:
• If the combinations are ordered, then “ab” and “ba” are two different combinations.
• If the combinations are unordered, then “ab” and “ba” are the same combination, because they consist of the same letters but in a different order.
• There will always be more ordered combinations.

Example: In how many ways can the letters “a” and “b” be combined?

If order matters and with replacement

Solution as a matrix

Solution using a tree diagram

Solution by calculation:

2·2 = 4 possibilities
As both the calculation, the matrix, and the tree diagram show, there are 4 possible combinations.

If order matters and without replacement

Solution using a matrix

Solution using a tree diagram

Solution by calculation:

2·1 = 2 possibilities
As both the calculation, the matrix, and the tree diagram show, there are 2 possible combinations.

If order does not matter and with replacement

Solution using a matrix

Solution using a tree diagram

Calculation:

The calculation is somewhat special and will be shown later.
As both the matrix and the tree diagram show, there are 3 possible combinations.

If order does not matter and without replacement

Calculation:

The calculation is somewhat special and will be shown later.
As both the matrix and the tree diagram show, there is 1 possible combination.