Consider Pascal's Triangle, as shown in the following diagram:
The expansion of the above binomial will have #n + 1# terms, in #(A + B)^n#. So, our binomial expansion will have #10 + 1 = 11 # terms.
We now search for the row in the triangle with 11 terms. This is the bottom-most row, with coefficients #1-10-45-...-45-10-1#. These are what we will multiply our descending powers of A with and our ascending powers of B with (in #(A + B)^n#, of course).
This may all seem like gibberish, but here is the actual thing:
#(x + y)^10 = 1 xx x^10 xx y^0 + 10 xx x^9 xx y^1 + 45 xx x^8 xx y^2 + 120 xx x^7 xx y^3 + 210 xx x^6 xx y^4 + 252 xx x^5 xx y^5 + 210 xx x^4 xx y^6 + 120 xx x^3 xx y^7 + 45 xx x^2 xx y^8 + 10 xx x xx y^9 + 1 xx x xx y^10#
#=>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 + 10xy^9 + y^10#
Hopefully this helps!
