Binomial theorem

What the symbols mean

$n$

the power the bracket is raised to (a whole number 0, 1, 2, …)

$k$

which term: the calculator gives the coefficient of $x^k$ in $(ax+b{)}^n$

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$

the binomial coefficient "n choose k" $=\frac{n!}{k!(n-k)!}$

$a$

in the calculator, the coefficient of x inside the bracket

$b$

in the calculator, the constant inside the bracket

$T$

the coefficient of $x^k$: $\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^k{b}^{n-k}$

When to use it

Use it to expand a bracket raised to a power, or to pick out one term ("find the coefficient of $x^3$ in $(2x-1{)}^7$") without writing the whole expansion. It also underlies the binomial probability distribution, where the same coefficients count arrangements.

Why it works

Multiplying out $(a+b{)}^n$ means choosing a or b from each of the n brackets. A term $a^{n-k}{b}^k$ arises once for every way of choosing which k brackets supply b, and there are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ such ways.

Common mistakes

• Forgetting to raise the coefficient: in $(2x+3{)}^5$ the $x^2$ term contains $(2x{)}^2=4x^2$, not $2x^2$.
• Dropping a negative sign: in $(x-1{)}^4$, b = −1, so the signs alternate.
• Mixing up which power goes with which: in $\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^k{b}^{n-k}$ the powers always add up to n.