Binomial Theorem — Statement and the General Term

The Binomial Theorem gives a direct formula for expanding $(x+y)^n$ for a positive integer $n$, without multiplying $(x+y)$ by itself $n$ times.

A few structural facts worth noting:

• The expansion has exactly $n+1$ terms.
• The powers of $x$ decrease from $n$ to $0$ while the powers of $y$ increase from $0$ to $n$; in every term the two exponents add up to $n$.
• The coefficients $^{n}C_{0}, \,^{n}C_{1}, \ldots, \,^{n}C_{n}$ are exactly the binomial coefficients from Pascal's triangle (previous section).
• Putting $x=y=1$ shows $2^n = \, ^{n}C_{0}+\,^{n}C_{1}+\cdots+\,^{n}C_{n}$ — the sum of a row of Pascal's triangle is always a power of 2.

The general term. The $(r+1)$th term of the expansion, counting from the first term as $T_1$, is

$T_{r+1} = \, ^{n}C_{r}\,x^{n-r}y^{r}$ …