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

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Nov 20, 2017

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${\left(x + y\right)}^{6} = {x}^{6} + 6 {x}^{5} y + 15 {x}^{4} {y}^{2} + 20 {x}^{3} {y}^{3} + 15 {x}^{2} {y}^{4} + 6 x {y}^{5} + {y}^{6}$

#### 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

${\left(x + y\right)}^{6} = {\text{^6C_0x^6+""^6C_1x^(6-1)y^1+""^6C_2x^(6-2)y^2+""^6C_3x^(6-3)y^3+""^6C_4x^(6-4)y^4+""^6C_5x^(6-5)y^5+}}^{6} {C}_{6} {y}^{6}$

$= {x}^{6} + 6 {x}^{5} y + 15 {x}^{4} {y}^{2} + 20 {x}^{3} {y}^{3} + 15 {x}^{2} {y}^{4} + 6 x {y}^{5} + {y}^{6}$

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