# Binomial Coefficients Calculator

An easy to use calculator that calculates the binomial coefficients ${n\choose k}$ from k= 0 to k = n included in the binomial theorem expansion.

## Use of the Binomial Coefficients Calculator

Enter the exponent $n$ as a positive integer greater than 1 and press "Expand". The outputs are the coefficients ${n\choose k}$ from k = 0 to k = n.

 n = 3

## Binomial Theorem Expansion and the Binomial Coefficients

The binomial coefficients ${n\choose k}$ that the above calculator compute are included in the binomial expansion theorem formula as follows.

$$(a x + b y)^n = \sum_{k=0}^{n} {n\choose k} (a x)^{n-k} (a y)^k$$

Example
$(2 x - 3 y)^3 = \sum_{k=0}^{3} {3\choose k} (2 x)^{3-k} (-3 y)^k$
$= {3\choose 0} (2 x)^{3-0} (-3 y)^0 + {3\choose 1} (2 x)^{3-1} (-3 y)^1 + {3\choose 2} (2 x)^{3-2} (-3 y)^2 + {3\choose 3} (2 x)^{3-3} (-3 y)^3$