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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

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Binomial Coefficient Calculator

Combination of k choose n \( n \choose k \) or \( C_{n}^{k} \)



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.

Answers to Questions

What is the binomial coefficient? (Definition)

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.

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

What are binomial coefficient properties?

The folowing formulas can be useful:

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

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

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

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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