top of page

Permutations & Combinations Calculator

Use this permutations and combinations calculator to count possible outcomes. Use it as a permutations calculator when order matters (e.g., exact finishes like first, second, and third place). Use it as a combinations calculator when order doesn't matter (e.g., parlay leg groups, lineups). Enter the total number of items and the number selected, and the calculator instantly returns both permutations and combinations.

Permutations vs Combinations: What's the Difference?

A permutation is an arrangement of items where the order of the items matters. For example, in exact finish markets, you might project which players will finish 1st, 2nd, and 3rd in a tournament. In this case, if the order changes, the outcome changes, so order "matters" and you're dealing with permutations.

​

A combination is an arrangement of items where the order of the items doesn't matter. For example, choosing possible starting lineups from a set of players is a combination because the order of the starters doesn't matter, they're just in the category of "starter" or "not starter". The order of the players doesn't change the group, so order "doesn't matter" and you're dealing with combinations.

​

Permutation formula

 

When order matters, use the permutation formula:

nPr = n! / (n - r)!

 

Where:

  • n = total number of items

  • r = number of items selected

  • ! = factorial

​​

Combination formula

 

When order doesn't matter, use the combination formula:

nCr = n! / (r!(n - r)!)

 

This removes duplicate arrangements, which is why combinations are always less than or equal to permutations for the same numbers.

​

Examples​

Permutation example: exact finish market

 

Suppose 10 golfers are in a tournament, and you want to know how many possible ways 3 specific finishing spots could be filled: 1st, 2nd, and 3rd.

​

Because finishing order matters, this is a permutation:

10P3 = 10 * 9 * 8 = 720

​

There are 720 possible exact finish orders.

​

Combination example: lineup combinations

 

Suppose you're choosing 3 players from a pool of 10 for a lineup build or betting scenario. The order of the players doesn't matter.

​

Because order doesn't matter, this is a combination:

10C3 = 120

​

There are 120 possible 3 player combinations.

​

Quick rule

 

Use permutations for sports betting situations like exact finishes, where changing the order changes the outcome.

Use combinations for sports betting situations like lineups or groups, where the same items still count as the same selection even if reordered.

Permutations & Combinations Calculator FAQ

What is a permutations and combinations calculator?

  • A permutations and combinations calculator shows how many possible outcomes can be created from a set of items. In sports betting, it can help you measure how many possible exact finish orders, parlay groupings, team combinations, or player combinations exist in a given scenario.

  • A permutations calculator is used when order matters.

  • A combinations calculator is used when order doesn't matter.

​​

What's the difference between permutations and combinations?

  • The difference is simple:

    • Permutations count outcomes where order matters

    • Combinations count outcomes where order doesn't matter

  • In sports betting, an exact finish market is a permutation because 1st, 2nd, and 3rd must be in the correct order. A group of teams in a parlay or a set of players in a build is a combination because the order of selection doesn't change the group.​​

​​

How do you calculate permutations?

  • Permutations are calculated with the formula:

    • nPr = n! / (n - r)!

    • Where:

      • n = total number of items

      • r = number of items selected

      • ! = factorial

  • This formula is used when order matters.

​​

How do you calculate combinations?

  • Combinations are calculated with the formula:

    • nCr = n! / (r!(n - r)!)

    • Where:

      • n = total number of items

      • r = number of items selected

      • ! = factorial

  • This formula is used when order does not matter.

​​

How does a combinations and permutations calculator help sports bettors?

  • A combinations and permutations calculator helps sports bettors quickly measure how many possible outcomes exist in a market or betting scenario. That can be useful for understanding complexity, comparing possible groupings, building models, and thinking more clearly about probability.

  • It's especially useful for markets involving exact finishes, grouped selections, lineup builds, playoff combinations, and other multi-outcome scenarios.

​​

What's an example of a permutation in sports betting?

  • A simple example is predicting the exact top 3 finishers in a race.

  • If you choose Golfer A to finish 1st, Golfer B to finish 2nd, and Golfer C to finish 3rd, that's different from choosing Golfer B 1st, Golfer A 2nd, and Golfer C 3rd. Because the order changes the outcome, it's a permutation.

​​

What's an example of a combination in sports betting?

  • A simple example is choosing 5 players to include in a starting lineup from a pool of 7 players.

  • If you choose Player A, B, C, D, and E, it's the same starting lineup as E, D, C, B, A. Because order doesn't matter, this is a combination.

Learn More About Betting Calculators

Check Out These Articles on a Range of Sports Betting Math Topics

bottom of page