Sometimes we need to consider whether the combinations we can create are ordered or unordered. If the combinations are ordered, the sequence matters. Conversely, if the combinations are unordered, the sequence does not matter.

That means:
• If the combinations are ordered, "ab" and "ba" are two different combinations.
• If the combinations are unordered, "ab" and "ba" are the same combination because they are the same letters in a different order.
• There will always be more ordered combinations than unordered ones.

Example: How many ways can the letters "a" and "b" be combined?

If order matters and with replacement

Solution as a matrix

Solution as a counting tree

Solution as a calculation:

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

If order matters and without replacement

Solution as a matrix

Solution as a counting tree

Solution as a calculation:

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

If order does not matter and with replacement

Solution as a matrix

Solution as a counting tree

Calculation:

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

If order does not matter and without replacement

Calculation:

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