Question

What is the fifth term of the expansion of `(a + b)^8`?

Answer 1

`70a^4 b^4`

Explanation:

The fifth term is the middle term of nine, with coefficient given by all the ways of choosing `4` items out of `8`, namely the ways of choosing `4` `a`'s out of `8` binomial factors.

`((8),(4)) a^4 b^4=(8!)/(4! 4!) a^4 b^4`

`=(8xx7xx6xx5)/(4xx3xx2xx1) a^4 b^4`

`=1680/24 a^4 b^4`

`= 70a^4b^4`

The coefficient `((8),(4)) = 70` can be picked out as the middle term of the row of Pascal's triangle that begins `1, 8,...`:

In full, we have:

`(a+b)^8 = sum_(k=0)^8 ((8),(k)) a^(8-k) b^k`

`=a^8+8a^7b+28a^6b^2+56a^5b^3+70a^4b^4+56a^3b^5+28a^2b^6+8ab^7+b^8`

which is just a particular example of the general Binomial Theorem:

`(a+b)^n = sum_(k=0)^n ((n),(k)) a^(n-k) b^k`

where `((n),(k)) = (n!)/((n-k)! k!)`