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.
Binomial Coefficient - dCode
Tag(s) : Combinatorics
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!
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.
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.
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: $$ (x+y)^{4} = x^4 + {4 \choose 1} x^3 y + {4 \choose 2} x^2 y^2 + {4 \choose 3} x y^3 + y^4 = x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4 $$
The folowing formulas can be useful for binomial coefficients:
$$ {n \choose k} = {n \choose n-k} $$
$$ {n \choose k} + {n \choose k+1} = {n+1 \choose k+1} $$
$$ {n \choose k} = {\frac{n}{k}}{n-1 \choose k-1} $$
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.
dCode retains ownership of the online 'Binomial Coefficient' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Binomial Coefficient download for offline use on PC, tablet, iPhone or Android !
Please, check our community Discord for help requests!