Combination Calculator
Enter a non-negative integer (maximum 170)
Enter a non-negative integer less than or equal to n
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Enter a non-negative integer
Result
Pascal's Triangle
In Pascal's triangle, each number is the sum of the two numbers above it. The value at position (n,r) represents nCr, the number of ways to choose r items from n items.
Enter the total number of items (e.g., 52 cards in a deck)
Enter the number of items drawn (e.g., 5 cards drawn from the deck)
Enter the number of target items (e.g., 13 hearts in a deck)
Enter the target count (e.g., exactly 2 hearts)
Probability Result
Combination Formulas
Standard Combination (nCr)
The number of ways to choose r items from a set of n distinct items, where the order does not matter.
Permutation (nPr)
The number of ways to arrange r items from a set of n distinct items, where the order matters.
Sum of Combinations
The sum of combinations from a starting value a to an ending value b.
Combination with Repetition
The number of ways to choose r items from a set of n distinct items, where repetition is allowed.
Permutation with Repetition
The number of ways to arrange r items from a set of n distinct items, where repetition is allowed.
Probability Using Combinations
The probability of drawing exactly x target items when k items are drawn from a set of n items that contains m target items:
Common Examples
Properties of Combinations
- Symmetry: nCr = nCn-r
- Sum of combinations in a row: Σr=0n nCr = 2n
- Pascal's Identity: nCr = n-1Cr-1 + n-1Cr