It's a bummer if they want you to do the formula for binomial expansion in each term.
But the other way is just use Pascal's triangle. This will look longer in explanation but once you get it, this is a lot faster.
(x+y)0 = 1
(x+y)1 = 1 1
(x+y)2 = 1 2 1
(x+y)3 = 1 3 3 1
(x+y)4 = 1 4 6 4 1
(x+y)5 = 1 5 10 10 5 1
(x+y)6 = 1 6 15 20 15 6 1
That was supposed to be written symmetrically to see that you should just add the adjacent values to get the value below. Example look at (x+y)2 = 1 2 1, you should add 1+2 to get the 3 in the (x+y)3.
So the constants for (x+y)6 = 1 6 15 20 15 6 1
Now how about the variables. The first one would just be exponent of the binomial and it's customary to start with x. so the variable of first term is x6. The next one is you just need to decrease the exponent of x by 1 while increasing that of y, meaning the variable of next term is x5y1 (even though we don't really put 1) then x4y2 up to y6.
If the question is x+y, then it's already done because you'll just multiply the constants with corresponding variables. However since it is x-y, you should be mindful that y is negative. Therefore all y variables that has odd exponents will yield a negative value. For example (-y)(-y)(-y) = -y3 while (-y)(-y) will be positive which is y2.
So considering everything, the answer should be:
(x-y)6 = x6 - 6x5y + 15x4y2 - 20x3y3 + 15x2y4 - 6xy5 + y6