The binomial theorem

We are now ready to prove the binomial theorem. We will give another proof later in the module using mathematical induction.

Theorem (Binomial theorem)

For each positive integer \(n\),

\[ (a+b)^n = a^n + \dbinom{n}{1}a^{n-1}b + \dbinom{n}{2}a^{n-2}b^2+\dots+\dbinom{n}{r}a^{n-r}b^r+\dots+ \dbinom{n}{n-1}ab^{n-1} + b^n. \]

Suppose that we have \(n\) factors each of which is \(a+b\). If we choose one letter from each of the factors of

\[ (a+b)(a+b)(a+b)\dotsm(a+b) \]

and multiply them all together, we obtain a term of the product. If we do this in every possible way, we will obtain all of the terms.


\((a+b)^n = a^n + \dbinom{n}{1}a^{n-1}b + \dbinom{n}{2}a^{n-2}b^2+\dots+\dbinom{n}{r}a^{n-r}b^r+\dots+ \dbinom{n}{n-1}ab^{n-1} + b^n\).


The binomial theorem can also be stated using summation notation:

\[ (a+b)^n = \sum_{r=0}^{n}\dbinom{n}{r}a^{n-r}b^r. \]

Substituting with \(a=1\) and \(b=x\) gives

\[ (1+x)^n = \dbinom{n}{0}+ \dbinom{n}{1}x + \dbinom{n}{2}x^2+\dots+\dbinom{n}{r}x^r+\dots+ \dbinom{n}{n-1}x^{n-1} + \dbinom{n}{n}x^n. \]

We can now display Pascal's triangle with the notation of the binomial theorem.

Pascal's triangle using the binomial theorem
\(n\) \(x^0\) \(x^1\) \(x^2\) \(x^3\) \(x^4\) \(x^5\) \(x^6\) \(x^7\) \(x^8\)
0 \(\dbinom{0}{0}\)
1 \(\dbinom{1}{0}\) \(\dbinom{1}{1}\)
2 \(\dbinom{2}{0}\) \(\dbinom{2}{1}\) \(\dbinom{2}{2}\)
3 \(\dbinom{3}{0}\) \(\dbinom{3}{1}\) \(\dbinom{3}{2}\) \(\dbinom{3}{3}\)
4 \(\dbinom{4}{0}\) \(\dbinom{4}{1}\) \(\dbinom{4}{2}\) \(\dbinom{4}{3}\) \(\dbinom{4}{4}\)
5 \(\dbinom{5}{0}\) \(\dbinom{5}{1}\) \(\dbinom{5}{2}\) \(\dbinom{5}{3}\) \(\dbinom{5}{4}\) \(\dbinom{5}{5}\)
6 \(\dbinom{6}{0}\) \(\dbinom{6}{1}\) \(\dbinom{6}{2}\) \(\dbinom{6}{3}\) \(\dbinom{6}{4}\) \(\dbinom{6}{5}\) \(\dbinom{6}{6}\)
7 \(\dbinom{7}{0}\) \(\dbinom{7}{1}\) \(\dbinom{7}{2}\) \(\dbinom{7}{3}\) \(\dbinom{7}{4}\) \(\dbinom{7}{5}\) \(\dbinom{7}{6}\) \(\dbinom{7}{7}\)
8 \(\dbinom{8}{0}\) \(\dbinom{8}{1}\) \(\dbinom{8}{2}\) \(\dbinom{8}{3}\) \(\dbinom{8}{4}\) \(\dbinom{8}{5}\) \(\dbinom{8}{6}\) \(\dbinom{8}{7}\) \(\dbinom{8}{8}\)

The \(n\)th row of this table gives the coefficients of \((1+x)^n\), where \(\dbinom{n}{r}\) is the coefficient of \(x^r\) in this expansion. Numbers of the form \(\smash{\dbinom{n}{r}}\) are called binomial coefficients.

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