Combination vs Permutation: What’s the Difference?

What Is a Combination?

A combination is a selection of items from a larger set where the order of selection does not matter. Think of it as choosing a group. If you pick apples, bananas, and cherries for a fruit salad, it does not matter which one you grabbed first. The salad is the same regardless.

The formula for combinations (also called the binomial coefficient) is:

C(n, r) = n! / (r! × (n − r)!)

Where n is the total number of items and r is how many you are choosing. The r! in the denominator is what eliminates duplicate arrangements.

Combination Example

A committee of 3 must be chosen from 10 candidates. How many different committees are possible?

C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120 committees

The committee of {Alice, Bob, Carol} is the same as {Carol, Alice, Bob} because the same three people serve regardless of the order they were picked.

What Is a Permutation?

A permutation is an arrangement of items where order matters. Think of it as ranking or sequencing. If you are awarding gold, silver, and bronze medals to 3 out of 10 athletes, who finishes first vs. third is a completely different outcome.

The formula for permutations is:

P(n, r) = n! / (n − r)!

Notice the only difference from the combination formula is the missing r! in the denominator. That is because permutations count every distinct ordering of each group.

Permutation Example

How many ways can 10 athletes finish in 1st, 2nd, and 3rd place?

P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720 arrangements

Notice: 720 is exactly 6 times 120. That is because each group of 3 people can be arranged in 3! = 6 ways (gold-silver-bronze orderings). This relationship — P(n,r) = C(n,r) × r! — always holds.

Key Differences Between Combinations and Permutations

The core distinction boils down to one word: order. But knowing that conceptually and applying it to real problems are different skills. Here are the practical differences:

• Size of results: Permutations always produce a larger count than combinations for the same n and r (except when r = 0 or r = 1, where they are equal).
• Formula difference: The only structural difference is the r! in the denominator. Combinations divide out the duplicate orderings; permutations do not.
• Problem keywords: Combination problems use words like “choose,” “select,” “group,” and “committee.” Permutation problems use “arrange,” “rank,” “order,” and “sequence.”
• With repetition: Both have “with repetition” variants. Combinations with repetition use the formula C(n+r-1, r). Permutations with repetition use n^r (each position can be any of n items).

When to Use Combinations

Use the combination formula whenever you are counting groups, subsets, or selections where rearranging the chosen items does not create a new outcome:

• Lottery odds: Powerball asks you to pick 5 numbers from 69. The order you write them does not matter — C(69, 5) = 11,238,513.
• Team selection: Choosing 5 starters from a 12-player roster — C(12, 5) = 792.
• Card hands: A 5-card poker hand from a 52-card deck — C(52, 5) = 2,598,960.
• Menu items: Picking 3 side dishes from 8 options.
• Exam questions: “Answer any 5 out of 8 questions” — C(8, 5) = 56 possible sets.

When to Use Permutations

Use the permutation formula whenever the arrangement, sequence, or ranking of chosen items matters:

• Race results: 1st, 2nd, 3rd from 20 runners — P(20, 3) = 6,840.
• PIN codes and passwords: A 4-digit PIN using digits 0-9 with no repeats — P(10, 4) = 5,040.
• Seating arrangements: Assigning 4 people to 4 labeled chairs — P(4, 4) = 4! = 24.
• Batting order: Arranging 9 baseball players in a lineup — P(9, 9) = 362,880.
• License plates: 3 letters followed by 4 digits (with repetition allowed) — 26³ × 10&sup4; = 175,760,000.

Common Mistakes to Avoid

Students mix these up constantly. Here are the traps:

• Defaulting to permutations: Most real-world counting problems are combinations. If you are unsure, ask: “If I rearrange my selection, do I get a different thing?” Usually the answer is no.
• Forgetting repetition: Standard formulas assume each item can be chosen only once. If items can repeat (like digits in a code), you need the “with repetition” variants.
• Confusing n and r: n is always the total pool. r is always how many you pick. Getting these backward produces wildly wrong answers.
• Not simplifying factorials: You almost never need to compute full factorials. Cancel common terms: 10! / 7! = 10 × 9 × 8.

The Bottom Line

Combinations and permutations are two sides of the same coin. Combinations count groups. Permutations count ordered arrangements. The formulas differ by a single factor of r!. When in doubt, ask one question: does order matter? That determines which formula to use.

Try your own calculations with our free combination calculator or permutation calculator.

Frequently Asked Questions