Search for a tool
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.

Results

Binomial Coefficient -

Tag(s) : Combinatorics

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Thanks to your feedback and relevant comments, dCode has developed the best 'Binomial Coefficient' tool, so feel free to write! Thank you!

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

### What are binomial coefficient properties?

The folowing formulas can be useful 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.

## Source code

dCode retains ownership of the online 'Binomial Coefficient' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Binomial Coefficient' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Binomial Coefficient' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Binomial Coefficient' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!