For a binomial expansion:
#(x+y)^n# we have:
#((n),(r))x^(n-r)y^r#
#sum_(r=0)^(n)((n),(r))x^(n-r)y^r#
Where:
#((n),(r))=color(white)(0)^n C_(r)=(n!)/(r!(n-r)!)#
Beginning with #r=0#
#((10),(0))x^10(-y)^0+((10),(1))x^9(-y)^1+((10),(2))x^8(-y)^2#
#((10),(3))x^7(-y)^3+((10),(4))x^6(-y)^4+((10),(5))x^5(-y)^5#
#((10),(6))x^4(-y)^6+((10),(7))x^3(-y)^7+((10),(8))x^2(-y)^8#
#((10),(9))x^1(-y)^9+((10),(10))x^0(-y)^10#
Next calculate #((n),(r))#
#(1)x^10(-y)^0+(10)x^9(-y)^1+(45)x^8(-y)^2#
#(120)x^7(-y)^3+(210)x^6(-y)^4+(252)x^5(-y)^5#
#(210)x^4(-y)^6+(120)x^3(-y)^7+(45)x^2(-y)^8#
#(1)x^1(-y)^9+(1)x^0(-y)^10#
Expand brackets. Remember to pay attention to the signs of #bby#
#x^10-10x^9y+45x^8y^2-120x^7y^3+210x^6y^4-252x^5y^5#
#+210x^4y^6-120x^3y^7+45x^2y^8-xy^9+y^10#
To make this easier, there are a couple of things worth remembering:
#color(white)(0)^nC_(r)=color(white)(0)^nC_(n-r)#
And #(-y)^n# is negative for odd powers and positive for even powers.
