Combinations Calculator (nCr)

combinations · nCr

number of ways to choose r items from a set of n items (order does not matter)
📐 formula
\[ C(n,r) = \binom{n}{r} = \frac{n!}{(n-r)! \; r!} \]

\( n \ge r \ge 0 \) both integers

enter values
56
\( \frac{8!}{5! \cdot 3!} \)
\[ C(8,3) = \frac{8!}{5! \cdot 3!} = 56 \]

understanding combinations

A combination is a selection of items from a set without regard to the order. The number of combinations of \( n \) distinct objects taken \( r \) at a time is denoted by \( C(n,r) \), \( \binom{n}{r} \) or \( {}_nC_r \).

example: choosing 3 items from a set of 8 gives \[ C(8,3) = \frac{8!}{5! \cdot 3!} = \frac{8\cdot7\cdot6}{3\cdot2\cdot1} = 56 \] possible groups.

when to use combinations

Use combinations when you want to count subgroups where order is irrelevant (e.g., lottery numbers, card hands, selecting a committee). For ordered arrangements, use permutations calculator.

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