Binomial Theorem

At a glance 11 formulas

1. 1General term$T_{r+1} = \binom{n}{r}\,a^{\,n-r}\,b^{\,r}$$\text{in } (a + b)^n,\ r = 0, 1, \dots, n \ (n + 1 \text{ terms})$
2. 2Middle term$n \text{ even: } T_{\frac{n}{2} + 1}$$n \text{ odd: } T_{\frac{n+1}{2}} \text{ and } T_{\frac{n+3}{2}}$
3. 3Symmetry$\binom{n}{r} = \binom{n}{n - r}$$\text{the largest } \binom{n}{r} \text{ is the middle one}$
4. 4Term independent of xWrite the power of $x$ in $T_{r+1}$ and set it to 0 (or to $k$ for the $x^k$ term). If no whole-number $r$ fits, there is no such term.
5. 5Sum of coefficients$\text{sum of coefficients of } f(x) = f(1)$$\binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{n} = 2^n$
6. 6Even and odd powers$\text{even powers of } f(x) = \dfrac{f(1) + f(-1)}{2}$$\text{odd powers} = \dfrac{f(1) - f(-1)}{2},\ \text{constant} = f(0)$
7. 7Weighted sum$\textstyle\sum_{r} r\binom{n}{r} = n\,2^{\,n-1}$$\text{since } r\binom{n}{r} = n\binom{n-1}{r-1}$
8. 8Neighbouring coefficients$\binom{n}{r} \div \binom{n}{r-1} = \dfrac{n - r + 1}{r}$
9. 9Combine before expandingMultiply brackets first when they pair up: $(1 + x)^n(1 - x)^n = (1 - x^2)^n$, so only even powers of $x$ survive.
10. 10Remainders$(1 + m)^n = 1 + nm + (\text{multiple of } m^2)$$\text{so } (1 + m)^n \div m^2 \text{ leaves what } 1 + nm \text{ leaves}$
11. 11Rational termsIn $(\sqrt[p]{A} + \sqrt[q]{B})^n$, with no perfect powers inside the roots, $T_{r+1}$ is rational exactly when $p$ divides $n - r$ and $q$ divides $r$.