We need to use Pascal's Triangle, shown in the picture below, for this expansion.
Because the binomial is raised to the #5th# power, we need to use the #5th# row of the triangle. The #5th# row is the one that features #color(red) (1,5,10,10,5,)# and #color(red)1#.
In the expansion, #color(navy)x# will be the first term and #color(green)(-y)# will be the second. Thus, the expression looks like this.
#(color(red)1*color(green)((-y)^5)*color(navy)(x^0))+(color(red)5*color(green)((-y)^4)*color(navy)(x^1))+(color(red)10*color(green)((-y)^3)*color(navy)(x^2))+
(color(red)10*color(green)((-y)^4)*color(navy)(x^3))+
(color(red)5*color(green)((-y)^5)*color(navy)(x^4))+(color(red)1*color(green)((-y)^0)*color(navy)(x^5))#
For each term from Pascal's Triangle, the exponent of the first term, #color(navy)x#, increases by #1#, while the exponent of the second term, #color(green)(-y)#, decreases by #1#.
Now, we can simplify and combine like terms.
#color(green)(-y^5)+color(red)5color(green)(y^4color(navy)(x))-color(red)(10)color(green)(y^3)color(navy)(x^2)+color(red)10color(green)(y^4)color(navy)(x^3)-color(red)(5)color(green)(y^5)color(navy)(x^4)+color(navy)(x^5)#
