The coefficients of the expansion are given by the 6th row of Pascal's triangle, where the top row is row zero, the next is row one, etc.
color(white)(AAAaaaAAAAAA)1
color(white)(AAAAaAAAaA)1color(white)(aa)1
color(white)(aaaaaaaa)1color(white)(aa)2color(white)(aa)1
color(white)(aaaaaa)1color(white)(aa)3color(white)(aaa)3color(white)(aa)1
color(white)(aaaa)1color(white)(aa)4color(white)(aaa)6color(white)(aaa)4color(white)(aa)1
color(white)(aa)1color(white)(aaa)5color(white)(aa)10color(white)(aa)10color(white)(aa)5color(white)(aa)1
1color(white)(aaa)6color(white)(aa)15color(white)(aa)20color(white)(aa)15color(white)(aa)6color(white)(aa)1
The coefficients of the 6th row are used because we are expanding to the 6th power. The coefficients are color(red)1, color(red)6, color(red)(15), color(red)(20), color(red)(15), color(red)6, color(red)1.
To expand (x-y)^6, use the coefficients in front of
x^6y^0, color(white)(aa)x^5y^1, color(white)(aa)x^4y^2, etc.,
with the exponent of x starting at 6 and decreasing by one in each term, and the exponent of y starting at 0 and increasing by one in each term. Note the sum of the exponents in each term is 6.
Also, starting with +, alternate the signs of each term because of the -y term in the original binomial. If the binomial to be expanded was (x+y), the signs would all be positive.
color(red)1x^6y^0-color(red)6x^5y^1+color(red)(15)x^4y^2-color(red)(20)x^3y^3+color(red)(15)x^2y^4-color(red)6x^1y^5+color(red)1x^0y^6
Simplifying gives
x^6-6x^5y+15x^4y^2-20x^3y^3+15x^2y^4-6xy^5+y^6
