Statement of the Binomial Theorem for Positive Integral Index

Building Up From Small Cases

We already know how to expand small positive-integer powers of a binomial $(a+b)$ by repeated multiplication:

Three patterns are visible in every one of these expansions.

Pattern (i) -- number of terms. The expansion of $(a+b)^n$ always has exactly $n+1$ terms: $(a+b)^2$ has 3 terms, $(a+b)^3$ has 4 terms, and so on.

Pattern (ii) -- exponents move in opposite directions. In successive terms, the power of $a$ falls by 1 at each step (starting at $n$ and ending at $0$) while the power of $b$ rises by 1 at each step (starting at $0$ and ending at $n$). In every single term, the two exponents add up to exactly $n$.

Pattern (iii) -- the coefficients are combinations. The coefficient of the term containing $a^{n-r}b^r$ is $^{n}C_r$. This can be checked directly: in $(a+b)^4$, the coefficient of $a^2b^2$ is $6$, and $^{4}C_2 = \dfrac{4!}{2!\,2!}=6$; the coefficient of $a^3b$ is $4$, and $^{4}C_1=4$.

The Binomial Theorem

Putting these three patterns together for a general positive integer $n$ gives the binomial theorem for a positive integral index:

Here $a$ and $b$ can be any two real numbers (or, more generally, any two quantities for which the usual laws of algebra hold), and $n$ is a positive integer. The numbers $^{n}C_0, {}^{n}C_1, \ldots, {}^{n}C_n$ that appear as coefficients are called the binomial coefficients; a full proof that this formula holds for every positive integer $n$ is given in the next section.

A convenient special case is $a=1,\ b=x$:

$(1+x)^n = {}^{n}C_0 + {}^{n}C_1x + {}^{n}C_2x^2 + \cdots + {}^{n}C_nx^n = \sum_{r=0}^{n}{}^{n}C_r\,x^r,$ …