Permutations and combinations
Permutations and Combinations
Permutations and Combinations is a topic in Probability & Statistics in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.13(A), which requires students to develop strategies to use permutations and combinations to solve contextual problems.
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.