Permutations and combinations help students count outcomes without listing every possibility. This cheat sheet covers the main counting rules used in algebra, probability, statistics, and discrete math. Students need it to decide when order matters, when repetition is allowed, and which formula matches a situation.
It is useful for homework, test review, and probability problems involving selections or arrangements.
The core ideas begin with the multiplication principle and factorial notation. Permutations count ordered arrangements, while combinations count unordered selections. Many problems can be solved by identifying the total number of items , the number chosen , and whether repeats are allowed.
When events involve several stages, add counts for separate cases and multiply counts for connected choices.
Key Facts
- The factorial of a positive integer is , and .
- The multiplication principle says that if one choice has options and the next has options, then the total number of ordered outcomes is .
- The number of permutations of distinct objects taken at a time is .
- The number of combinations of distinct objects taken at a time is .
- If order matters and repetition is allowed, the number of length arrangements from choices is .
- If order does not matter and repetition is allowed, the number of selections is .
- The number of distinct arrangements of objects with repeated groups of sizes , , and is .
- For combinations, symmetry gives , so choosing items is equivalent to leaving out items.
Vocabulary
- Factorial
- A factorial, written , is the product of all positive integers from down to , with .
- Permutation
- A permutation is an arrangement where the order of the selected items matters.
- Combination
- A combination is a selection where the order of the selected items does not matter.
- Repetition
- Repetition means an item may be chosen more than once in the same counting process.
- Multiplication Principle
- The multiplication principle states that choices made in stages are counted by multiplying the number of options at each stage.
- Binomial Coefficient
- A binomial coefficient, written , counts the number of ways to choose items from items without order.
Common Mistakes to Avoid
- Using permutations when order does not matter is wrong because it counts the same group multiple times, such as counting and as different teams.
- Using combinations when order matters is wrong because positions or rankings create different outcomes, such as st, nd, and rd place winners.
- Forgetting the value is wrong because formulas like and depend on it.
- Allowing repetition when the problem says items are distinct or cannot be reused is wrong because formulas like assume every choice remains available each time.
- Adding when choices happen together is wrong because staged choices use multiplication; for example, shirts and pants make outfits, not .
Practice Questions
- 1 How many ways can students line up in a row?
- 2 A club has members. How many different committees of members can be chosen?
- 3 A password uses letters chosen from letters, and repetition is allowed. How many passwords are possible?
- 4 A teacher chooses students for a committee and also ranks students for first, second, and third prize. Explain why one situation uses combinations and the other uses permutations.
Understanding Permutations & Combinations Master Reference
A useful way to classify a counting problem is to imagine changing the positions of the chosen objects. If that change creates a new outcome, the problem is about arrangements. A race medal order, a password, and seating students in numbered chairs all have positions that matter.
A committee, a pizza topping group, and a set of lottery numbers do not have positions. This simple position test prevents the most common error, which is using an arrangement count for a selection problem.
Factorials appear because filling positions usually reduces the choices left for later positions. For example, arranging several different books on a shelf starts with many choices for the first space, then one fewer for the next space. Writing the product as a factorial saves work.
In calculations, avoid expanding every factorial immediately. Cancel matching factors first.
This makes large expressions manageable and reduces calculator mistakes. The value of zero factorial may seem strange, but it keeps counting rules consistent when no objects are chosen or when every object has been placed.
Repetition changes the story. A four digit access code can use the same digit more than once, so each position has the full set of digit choices. By contrast, drawing cards from a deck without replacement removes a card after it is drawn.
Repeated objects need special care too. The letters in a word such as BALLOON are not all unique.
Swapping one L with the other L produces no visible new arrangement. Dividing by the factorials of repeated groups removes these duplicate counts.
Many real problems have restrictions that standard formulas do not handle by themselves. Suppose two friends must sit together, or two particular students cannot be on the same team. Treat required objects as one block when they stay together.
For forbidden outcomes, count all possible outcomes first, then subtract the outcomes that break the rule. When a problem separates into nonoverlapping cases, count each case and combine the results. A careful list of cases matters because overlapping cases can accidentally count the same outcome twice.
Counting is closely connected to probability. Once every outcome is equally likely, probability equals the number of favorable outcomes divided by the number of possible outcomes. The counting method must match the way outcomes are generated.
A hand of cards is a selection, while the order of cards dealt is an arrangement. Students should write what one outcome means before choosing a method.
They should check whether the answer is reasonable by testing a small version of the situation. Small cases often reveal missing restrictions, unwanted repeats, or order that was counted more than once.