Permutations and combinations are both counting methods in statistics and probability, but they answer different kinds of questions. A permutation counts how many arrangements are possible when order matters. A combination counts how many selections are possible when order does not matter.
Knowing which method to use helps you solve problems about rankings, teams, passwords, schedules, and probability.
The key difference is whether changing the order creates a new outcome. For example, choosing gold, silver, and bronze medals is a permutation because first, second, and third are different roles. Choosing 3 students for a committee is a combination because the group stays the same no matter how you list the names.
The formulas are closely related, since combinations can be found by dividing permutations by the number of ways to reorder the chosen items.
Understanding Permutations vs Combinations
A reliable way to begin is to describe one outcome in words before doing any calculation. Write down the positions or roles that must be filled. If there are three seats labeled captain, vice captain, and treasurer, each seat is a separate decision.
With eight candidates, there are eight choices for captain, then seven remaining choices for vice captain, then six for treasurer. The total comes from multiplying eight times seven times six.
This step by step multiplication is called the multiplication principle. It often makes more sense than memorising a formula, especially when a problem has unusual rules.
The division used for unordered selections has a clear purpose. Imagine selecting three books from eight books. If you temporarily place the chosen books into first, second, and third positions, every group has been counted more than once.
The group containing A, B, and C can be listed in six orders. These are A B C, A C B, B A C, B C A, C A B, and C B A. They all describe one group, not six different groups.
Dividing by the number of rearrangements removes this repeated counting. For a selection of four items, each group is counted twenty four times before division.
Real probability problems depend on defining the outcomes correctly. A lottery ticket that matches a set of drawn numbers uses unordered selections because the drawing order usually does not change the ticket. A lock code uses ordered sequences because entering 1234 differs from entering 4321.
In card games, a five card hand is normally an unordered selection. A sequence of cards dealt one at a time can be ordered when the timing matters.
If all outcomes are equally likely, probability equals the number of favourable outcomes divided by the number of possible outcomes. Using the wrong counting method changes both parts of that fraction and can give a very misleading probability.
Pay close attention to words that signal restrictions. Phrases such as no repeats, at least one, adjacent, together, or cannot be chosen change the counting plan. A password may allow repeated characters, so each position can have the same number of choices.
Choosing students for a team usually does not allow one student to be selected twice. Problems with identical objects need extra care because swapping two identical items does not create a new arrangement. A useful habit is to test a small case by listing outcomes.
If your method says there are six results but you can only list three distinct results, some outcomes have been counted repeatedly. This quick check builds judgment that formulas alone cannot provide.
Key Facts
- Permutation means order matters, combination means order does not matter.
- Number of permutations of distinct objects taken at a time:
- Number of combinations of distinct objects taken at a time:
- Relationship between them:
- Factorial definition: , and
- For the same n and r, permutations are greater than or equal to combinations because ordering creates more outcomes.
Vocabulary
- Permutation
- A permutation is an arrangement of objects where the order of the objects matters.
- Combination
- A combination is a selection of objects where the order of the objects does not matter.
- Factorial
- A factorial is the product of all positive integers from a number down to 1, written with an exclamation mark.
- Distinct objects
- Distinct objects are items that are different from one another and can be told apart.
- Sample space
- A sample space is the full set of all possible outcomes in a counting or probability problem.
Common Mistakes to Avoid
- Using combinations when order matters, which gives too few outcomes because different arrangements are being treated as the same.
- Using permutations when order does not matter, which gives too many outcomes because the same group is counted multiple times in different orders.
- Forgetting the r! in the combination formula, which means you do not remove repeated orderings of the same selection.
- Making factorial errors such as thinking 0! = 0 or canceling terms incorrectly, which leads to wrong numerical answers even if the correct formula was chosen.
Practice Questions
- 1 How many different 4-letter arrangements can be made from the letters A, B, C, D, E if no letter is repeated?
- 2 A class of 10 students must choose 3 students to serve on a committee. How many different committees are possible?
- 3 A school is choosing 2 students from 8 to attend a workshop. Explain whether this situation should be solved with permutations or combinations, and state why.