Permutations and combinations
Permutations and Combinations
Permutations and Combinations is a topic in Probability & Statistics in the California Common Core State Standards. It is aligned to Standard S-CP.9, which requires students to use permutations and combinations to compute probabilities of compound events.
Permutations count ordered arrangements \(\left(\dfrac{n!}{(n-r)!}\right)\) and combinations count unordered selections \(\left(\dfrac{n!}{r!\,(n-r)!}\right)\).
Theory
To count choices of \(r\) items from \(n\):
- Permutation (order matters): \(_nP_r=\dfrac{n!}{(n-r)!}\).
- Combination (order does not): \(_nC_r=\dfrac{n!}{r!\,(n-r)!}\).
The counting formulas:
How to count arrangements or selections
- Decide if order matters (permutation) or not (combination).
- Identify \(n\) (total) and \(r\) (chosen).
- Apply \(_nP_r\) or \(_nC_r\).
- Simplify the factorials by cancelling.
Order matters, so use \(_5P_3\).
| \(_5P_3\) | \(=\) | \(\dfrac{5!}{(5-3)!}=\dfrac{5!}{2!}\) |
| \(=\) | \(5\cdot4\cdot3=60\) |
Order does not matter, so use \(_5C_3\).
| \(_5C_3\) | \(=\) | \(\dfrac{5!}{3!\,2!}\) |
| \(=\) | \(\dfrac{120}{6\cdot2}=10\) |
A committee is unordered, so use \(_8C_4\).
| \(_8C_4\) | \(=\) | \(\dfrac{8!}{4!\,4!}\) |
| \(=\) | \(\dfrac{40320}{24\cdot24}=70\) |
No — order matters for a lock, so it is really a permutation, not a combination.
Common pitfalls
Frequently asked questions
What is the difference between a permutation and a combination?
A permutation counts arrangements where order matters; a combination counts selections where order does not.
What is the permutation formula?
\(_nP_r=\dfrac{n!}{(n-r)!}\).
What is the combination formula?
\(_nC_r=\dfrac{n!}{r!\,(n-r)!}\).
How do you decide which to use?
Ask whether order matters: if yes, permutation; if no, combination.