Permutations vs Combinations

When Order Matters and When It Does Not

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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.

Key Facts

Permutation means order matters, combination means order does not matter.
Number of permutations of $n$ distinct objects taken $r$ at a time: $nPr = \frac{n!}{(n - r)!}$
Number of combinations of $n$ distinct objects taken $r$ at a time: $nCr = \frac{n!}{r!(n - r)!}$
Relationship between them: $nCr = \frac{nPr}{r!}$
Factorial definition: $n! = n(n - 1)(n - 2)\ldots(2)(1)$ , and $0! = 1$
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.

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