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Binomial Coefficient

Tool to calculate the values of the binomial coefficient (combination choose operator) used for the development of the binomial but also for probabilities and counting.

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Binomial Coefficient -

Tag(s) : Combinatorics

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# Binomial Coefficient

## Binomial Coefficient Calculator

### What is the binomial coefficient? (Definition)

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

With $n!$ the factorial of n.

### How to calculate a binomial coefficient?

The binomial coefficient uses factorial functions whose values are simplified:

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

### Why is it called the coefficient binomial?

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

### How to calculate the coefficient binomial with Pascal's triangle?

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).

### What are binomial coefficient properties?

The following formulas can be useful for binomial coefficients:

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

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

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

### When to use the binomial coefficient?

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.

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