Question

How do you use the Binomial Theorem to expand `(x + y) ^6`?

Answer 1

`(x+y)^6=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6`

Explanation:

The Binomial Theorem gives a time efficient way to expand binomials raised to a power and may be stated as

`(x+y)^n=sum_(r=0)^n""^nC_rx^(n-r)y^r`,

where the combination `""^nC_r=(n!)/((n-r)!r!)`.

So in this particular case we get

`(x+y)^6=""^6C_0x^6+""^6C_1x^(6-1)y^1+""^6C_2x^(6-2)y^2+""^6C_3x^(6-3)y^3+""^6C_4x^(6-4)y^4+""^6C_5x^(6-5)y^5+""^6C_6y^6`

`=x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6`