Binomial Coefficient Calculator
Calculate binomial coefficients C(n,k) with Pascal's triangle visualization and properties
Calculate C(n,k) = "n choose k"
Résultats
Entrez les valeurs et cliquez sur Calculer pour voir le résultat.
Theory & Formula
Theory
Binomial coefficients represent the number of ways to choose k items from n items without regard to order. They appear in Pascal's triangle and the binomial theorem.
Formula
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
The number of k-combinations from a set of n elements
Properties
Symmetry: \(\binom{n}{k} = \binom{n}{n-k}\)
Pascal's Identity: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\)
Sum of Row: \(\sum_{k=0}^{n} \binom{n}{k} = 2^n\)
Base Cases: \(\binom{n}{0} = \binom{n}{n} = 1\)
Binomial Theorem
\((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
Binomial coefficients are the coefficients in the expansion of (a+b)ⁿ
Example
How many ways can you choose 2 items from 5 items?
\(\binom{5}{2} = \frac{5!}{2! \times 3!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10\)