Binomial expansion | Mathematico

Expanding something like

(a + b)^{5}

would be possible, but boring. Consider

(a + b) (a + b) (a + b) (a + b) (a + b)

Terms of the form, for example,

a^{3} b^{2}

are obtained by choosing a $b$ from $2$ of the brackets (and thus an $a$ from the remaining $3$ brackets). So $a^{3} b^{2}$ would turn up

5 C 3 = (\binom{5}{3})

times.

General formula

Generally, we have

(a + b)^{n} = (\binom{n}{0}) a^{n} + (\binom{n}{1}) a^{n - 1} b + (\binom{n}{2}) a^{n - 2} b^{2} + \dots + (\binom{n}{n - 1}) a b^{n - 1} + (\binom{n}{n}) b^{n}

This formula looks pretty scary, but it makes a lot more sense once you've seen some examples.