Choose 5 from 10
\(\binom{10}{5} = \frac{10!}{5!\,5!} = 252\)
Binomial Coefficient Calculator
Calculate binomial coefficients C(n,k) with Pascal's triangle visualization and properties
Explore C(n, k)
Drag n and k. The Pascal-triangle row for n is shown below; the k-th cell is highlighted in green.
Quick presets
5
060
2
05
Pascal's triangle, row 5
C(5, 0)
1
C(5, 1)
5
C(5, 2)
10
C(5, 3)
10
C(5, 4)
5
C(5, 5)
1
Predict what will happen
Why is C(n, k) always equal to C(n, n−k)?
Try n = 8 and switch k between 3 and 5.
Symmetry
C(n, k) = C(n, n−k). The Pascal-triangle row mirrors around its center.
Common mistake
Order matters: C(n, k) counts unordered combinations. If order matters, use permutations P(n, k) = n! / (n−k)!.
Why it works
There are n! ways to arrange all n items in a row. Dividing by k! (for the chosen ones) and (n−k)! (for the rest) removes the orderings that don't change which items were picked, leaving the unordered count.
Results
Final Answer
\(\binom{5}{2} = 10\)
Step-by-step Solution
- Calculate binomial coefficient \(\binom{5}{2}\)
- Formula \(\binom{n}{k} = \frac{n!}{k!\,(n-k)!}\)
- Substitute values \(\binom{5}{2} = \frac{5!}{2!\,3!} = 10\)
Binomial coefficient
C(n, k) — read 'n choose k' — is the number of ways to choose k unordered items from a set of n. It also gives the coefficient of x^k in the expansion (1 + x)^n.
Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\)
Pascal's rule: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
Worked Examples
Lottery (6 of 49)
\(\binom{49}{6} = 13{,}983{,}816\)