Binomial Coefficient

The binomial coefficient is a number that represents the number of ways to choose $ k $ elements from $ n $ distinct elements, regardless of order. In other words, it measures the number of possible combinations (counting).

The binomial coefficient is noted $ {n \choose k} $ or $ C_{n}^{k} $ is read $ n $ choose $ k $ (or $ k $ among $ n $). Generally $ n $ is the total number of elements and $ k $ is the number of chosen elements.

The binomial coefficient and is defined by the formula $$ {n \choose k} = \frac{n!}{k!(n-k)!} $$ with $ n! $ the factorial of n.

In practice, factorials have values that simplify.

The values of the binomial coefficient appear in the development of the Newton binomial:

$$ (a+b)^{n}=\sum_{k=0}^{n}{n \choose k}a^{{n-k}}b^{k} $$

The value of the binomial coefficient $$ \binom{A}{B} $$ is found in Pascal's triangle at row A, column column B (in row and column are 0-indexed).

The following formulas can be useful for binomial coefficients:

$$ {n \choose k} = {n \choose n-k} $$

$$ {n \choose k} + {n \choose k+1} = {n+1 \choose k+1} $$

$$ {n \choose k} = {\frac{n}{k}}{n-1 \choose k-1} $$

$$ {n \choose 0} = 1 $$

$$ {n \choose n} = 1 $$

The binomial coefficient is used primarily in count and probability calculations. This is the basis for calculating the number of combinations of k elements out of n.