The Fundamental Counting Principle
Before diving into permutations and combinations, you need to understand the fundamental counting principle.
m × n ways. This extends to any number of events — just multiply.For example, if you have 4 shirts and 3 pants, the total number of outfits is 4 × 3 = 12.
If you also choose from 2 pairs of shoes, the total becomes 4 × 3 × 2 = 24.
24 outfitsThis principle is the foundation of all counting problems. Permutations and combinations are simply structured ways to apply it when you are selecting items from a set.
Why factorials appear
The counting principle also explains why factorials appear in the formulas: when arranging n distinct objects in a line, the first position has n choices, the second has n − 1, the third has n − 2, and so on, giving n! = n × (n − 1) × … × 1 total arrangements.
What Are Permutations (nPr)?
A permutation is an arrangement of objects where order matters. Choosing president, vice-president, and secretary from a club of 10 people is a permutation problem because the same three people assigned to different roles count as different outcomes.
nPr = n! / (n − r)!.For the club example: 10P3 = 10! / 7! = 10 × 9 × 8 = 720 possible officer assignments.
Intuitively, you have 10 choices for president, then 9 remaining choices for vice-president, then 8 for secretary.
The factorial in the denominator cancels the tail end of n! that you do not use.
720nPn = n! (arrange all items) and nP1 = n (just pick one).A useful tip from Solver AI: when a problem says "arrange," "order," "rank," or "assign to distinct positions," it is almost always a permutation problem.
What Are Combinations (nCr)?
A combination is a selection of objects where order does not matter. Choosing a committee of 3 members from a group of 10 — with no assigned roles — is a combination problem because the group {Alice, Bob, Carol} is the same as {Carol, Alice, Bob}.
nCr = n! / (r!(n − r)!).For our example: 10C3 = 10! / (3! × 7!) = 720 / 6 = 120.
Notice that nCr = nPr / r!. The r! in the denominator removes the duplicate arrangements of the same group.
120This is the key insight: combinations divide out the orderings that permutations count separately.
Notation and properties
The notation nCr is also written as C(n, r) or the binomial coefficient (n choose r).
nC0 = 1, nCn = 1, and the symmetry rule nCr = nC(n − r). These properties often simplify calculations — choosing 8 items from 10 is the same count as choosing 2 items from 10.Key Difference: When Does Order Matter?
The single question that determines whether you use permutations or combinations is: does the order of selection matter? If rearranging the same items creates a different outcome, use permutations. If rearranging changes nothing, use combinations.
Here are reliable tests:
| Type | Clues | Example | Why |
|---|---|---|---|
| Permutation | Rankings, assigned positions, passwords, phone numbers, seating arrangements, or any scenario where sequence affects the result. | Selecting a 4-digit PIN from digits 0–9 | 1234 is different from 4321. |
| Combination | Groups, teams, committees, subsets, or selections where members are interchangeable. | Choosing 5 cards from a deck of 52 | The hand {A♠, K♥, Q♦, J♣, 10♠} is the same regardless of the order you drew the cards. |
When in doubt, ask yourself: "If I swap two of my chosen items, do I get a different outcome?" If yes, use permutations. If no, use combinations.
Solver AI automatically detects which formula applies and shows you the reasoning.
Worked Examples
How many 3-letter "words" can be formed from the letters A, B, C, D, E with no repeats?
Order matters (ABC ≠ CAB), so use permutations: 5P3 = 5 × 4 × 3 = 60.
60A pizza shop offers 8 toppings. How many ways can you choose 3?
Order does not matter, so use combinations: 8C3 = 8! / (3! × 5!) = 56.
56From 12 students, a teacher selects a team captain, assistant captain, and note-taker.
Roles are distinct, so use permutations: 12P3 = 12 × 11 × 10 = 1320.
1320A lottery requires choosing 6 numbers from 1 to 49.
Order does not matter: 49C6 = 49! / (6! × 43!) = 13,983,816.
This enormous number explains why jackpot odds are so slim.
13,983,816Notice how the same set of items (numbers 1–49) yields vastly different counts depending on whether order matters. Always identify the type before reaching for a formula.
Common Mistakes and Tips
Counting problems trip students up more than almost any other topic. Here are the most frequent errors and how to avoid them.
Confusing permutations and combinations is the number-one error. Always determine whether order matters before calculating.
If a problem says "at least 2 women must be on the committee," break it into cases (exactly 2 women, exactly 3 women, etc.) and add the combinations for each case.
When a problem has overlapping categories, use the inclusion-exclusion principle or carefully define non-overlapping cases.
Standard permutation and combination formulas assume without replacement. If items can repeat, you need different formulas: n^r for ordered selections with replacement, or (n + r − 1)C r for unordered selections with replacement.
Tip: For complex problems, start by writing out small cases by hand to build intuition before applying formulas. Solver AI can verify your counting logic and catch errors in multi-step problems.