Combination Calculator (nCr)
Calculate combinations (nCr) instantly. Find how many ways to choose r items from n items when order doesn't matter. Free combination calculator.
Formula
C(10, 3) = 10! / (3! × (10 - 3)!)
Number of Combinations
120
Calculation Steps
10! = 3628800
3! = 6
(10 - 3)! = 5040
Explanation: There are 120 ways to choose 3 items from 10 items when order doesn't matter.
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Last updated: January 2026
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Frequently Asked Questions
What is the difference between combinations and permutations?
Combinations count selections where order doesn't matter—choosing cards A, B, C is the same as C, B, A. Permutations count arrangements where order matters—ABC, ACB, BAC, BCA, CAB, CBA are 6 different permutations. For n items taken r at a time: combinations = n!/(r!(n-r)!), permutations = n!/(n-r)!. Permutations are always ≥ combinations. Use combinations for teams, committees, lottery numbers; use permutations for rankings, PIN codes, race results.
How do I calculate nCr (n choose r)?
The formula is nCr = n! / (r! × (n-r)!). For example, 5C3 (choosing 3 from 5): 5! / (3! × 2!) = 120 / (6 × 2) = 10. Shortcut: write the top r numbers descending and divide by r!. So 5C3 = (5×4×3) / (3×2×1) = 60/6 = 10. Key properties: nC0 = nCn = 1, nC1 = n, and nCr = nC(n-r). This calculator handles the factorial math automatically.
What are real-world examples of combinations?
Lottery: Choosing 6 numbers from 49 = 49C6 = 13,983,816 combinations (that's why jackpots are rare). Poker hands: 5 cards from 52 = 52C5 = 2,598,960 possible hands. Team selection: Choosing 5 starters from 12 players = 12C5 = 792 ways. Pizza toppings: Selecting 3 from 10 toppings = 10C3 = 120 combinations. Committee formation: Choosing 4 members from 20 candidates = 20C4 = 4,845 possible committees.
Why is 0! equal to 1?
By convention, 0! = 1. This makes the combination formula work correctly: nCn = n! / (n! × 0!) = 1 (choosing all items gives exactly 1 way). It also preserves the pattern n! = n × (n-1)!, so 1! = 1 × 0! means 0! = 1. Mathematically, 0! counts the ways to arrange zero items—there's exactly one way to do nothing. This convention appears throughout mathematics in series expansions, probability, and counting problems.
How do I know when to use combinations instead of permutations?
Ask: 'Does the order of selection matter?' If selecting committee members where all have equal roles → combinations (order doesn't matter). If assigning President, VP, Secretary → permutations (order matters). If choosing lottery numbers → combinations (1-2-3 wins same as 3-1-2). If creating a PIN code → permutations (123 differs from 321). Another test: would swapping two selected items give a 'different' result? If yes, use permutations. If no, use combinations.