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
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!


Feedback and suggestions are welcome so that dCode offers the best 'Binomial Coefficient' tool for free! Thank you!

Binomial Coefficient

Binomial Coefficient Calculator

Combination of k choose n $ n \choose k $ or $ C_{n}^{k} $




Find $ n $ and $ k $ from a value


Answers to Questions (FAQ)

What is the binomial coefficient? (Definition)

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.

How to calculate a binomial coefficient?

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.

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:

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

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

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

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

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

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 "Binomial Coefficient" source code. Except explicit open source licence (indicated Creative Commons / free), the "Binomial Coefficient" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Binomial Coefficient" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Binomial Coefficient" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Binomial Coefficient" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Binomial Coefficient on dCode.fr [online website], retrieved on 2024-12-02, https://www.dcode.fr/binomial-coefficient

Need Help ?

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

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Binomial Coefficient' tool for free! Thank you!


https://www.dcode.fr/binomial-coefficient
© 2024 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
 
Feedback