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.
∀Aaaa∀∀∀1
∀∀a∀AaA1aa1
aaaaaaaa1aa2aa1
aaaaaa1aa3aaa3aa1
aaaa1aa4aaa6aaa4aa1
aa1aaa5aa10aa10aa5aa1
1aaa6aa15aa20aa15aa6aa1
The coefficients of the 6th row are used because we are expanding to the 6th power. The coefficients are 1, 6, 15, 20, 15, 6, 1.
To expand (x−y)6, use the coefficients in front of
x6y0, aax5y1, aax4y2, 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.
1x6y0−6x5y1+15x4y2−20x3y3+15x2y4−6x1y5+1x0y6
Simplifying gives
x6−6x5y+15x4y2−20x3y3+15x2y4−6xy5+y6
