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Discrete Mathematics · Quick Reference
Combinatorics Cheat Sheet
In one line: combinatorics is the mathematics of counting — figuring out how many ways something can happen without necessarily listing every possibility.
Key Ideas
1Multiplication Principle. If one task can be done in m ways and a second independent task in n ways, both together can be done in m*n ways.
2Permutations. Arrangements where order matters. The number of ways to arrange r items from n distinct items is P(n,r) = n! / (n-r)!.
3Combinations. Selections where order does NOT matter. The number of ways to choose r items from n distinct items is C(n,r) = n! / [r!(n-r)!].
4Factorial Notation. n! (n factorial) means n*(n-1)*(n-2)*...*2*1. By definition, 0! = 1.
5When to Use Permutations vs. Combinations. Use permutations when arrangement/order matters (like race rankings); use combinations when only the selection matters (like choosing a committee).
Worked Examples
How many ways can 3 runners finish 1st, 2nd, and 3rd out of 8 racers?
336 ways
How many ways can a committee of 3 be chosen from 8 people?
56 ways
A restaurant has 4 appetizers and 6 main courses. How many appetizer-main course combinations are possible?
24 combinations
Formulas
P(n,r) = n! / (n-r)!
C(n,r) = n! / [r!(n-r)!]
Practice Yourself
Find P(5,2).
20
Find C(5,2).
10
Find 5!.
120