Binomial Coefficient

The binomial coefficient is noted \({n \choose k} \) or \( C_{n}^{k} \) and is defined by the formula $$ {n \choose k} = \frac{n!}{k!(n-k)!} $$

With \( n! \) the factorial of n.

The binomial coefficient uses factorial functions whose values are simplified:

Example: \( {10 \choose 6} = \frac{10!}{6!4!} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 4 \times 3 \times 2 \times 1} = \frac{10 \times 9 \times 8 \times 7 }{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \)

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

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

Example: $$ (x+y)^{4} = x^4 + {4 \choose 1} x^3 y + {4 \choose 2} x^2 y^2 + {4 \choose 1} x y^3 + y^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4 $$

The folowing formulas can be useful:

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

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

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

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.

Example: The number of lotto combinations is 5 out of 49 ie \( {49 \choose 5} = 1906884 \) possible combinations.