Binomial Coefficients Calculator
An easy to use calculator that calculates the binomial coefficients \( \displaystyle {n\choose k} \) 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 "Calculate". The outputs are the coefficients\( \displaystyle {n\choose k} \) from \( k= 0 \) to \( k = n \).
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
\[(2 x - 3 y)^3 = \sum_{k=0}^{3} {3\choose k} (2 x)^{3-k} (-3 y)^k\]
\[
\begin{aligned}
= {}& {3\choose 0} (2x)^{3}(-3y)^0 \\
&+ {3\choose 1} (2x)^{2}(-3y)^1 \\
&+ {3\choose 2} (2x)^{1}(-3y)^2 \\
&+ {3\choose 3} (2x)^{0}(-3y)^3
\end{aligned}
\]
Use the values of \( {3\choose 0} = 1 \) , \( {3\choose 1} = 3 \) , \( {3\choose 2} = 3\) and \( {3\choose 3} = 1\)
\[ (2 x - 3 y)^3 = (2 x)^{3-0} (-3 y)^0 + 3 (2 x)^{3-1} (-3 y)^1 + 3 (2 x)^{3-2} (-3 y)^2 + (2 x)^{3-3} (-3 y)^3 \]
\[
\begin{aligned}
(2x-3y)^3 = {}&
(2x)^3(-3y)^0 \\
&+ 3(2x)^2(-3y)^1 \\
&+ 3(2x)^1(-3y)^2 \\
&+ (2x)^0(-3y)^3
\end{aligned}
\]
More References and links
Binomial Theorem Expansion Calculators.
Maths Calculators and Solvers.