The Binomial Expansion

A binomial has two terms, such as $a+b$ or $2-3x$. Expanding a power means multiplying its brackets and collecting like terms.

In $(a+b)^5$, each product chooses one term from each of five brackets. To make $a^3b^2$, choose $b$ twice and $a$ three times.

A coefficient is the factor multiplying the stated power. In $10a^3b^2$, the coefficient of $a^3b^2$ is 10. The symbol $\binom nk$ or ${}^nC_k$, read “n choose k”, counts ways to select $k$ brackets from $n$.

Use your calculator’s nCr function: enter $5\ {}^nC_r\ 2$ to get 10. The key may be labelled nCr or $\binom nr$. It calculates the coefficient count, not the constants, powers or signs of the whole term.

To understand the written formula, factorial means multiplying down to 1: $5!=5\times4\times3\times2\times1$. We define $0!=1$.

For two choices from five, there are $5\times4$ ordered choices. Divide by $2!=2$ because choosing the same two brackets in reverse order is not a new selection. Thus $\binom52=5\times4/2=10$.

For $k$ choices, dividing $n!$ by $(n-k)!$ leaves just the first $k$ factors. Divide by $k!$ because the same selected brackets can be chosen in $k!$ different orders:

MF19 supplies the coefficient formula and binomial expansion. For P1 the power $n$ is a positive integer. Taking $k=0,1,\ldots,n$ gives each term:

The two chosen powers add to $n$ because exactly $n$ brackets contribute. Choose none or all of them in only one way: $\binom n0=\binom nn=1$.