[tex]\bf (5x-y)^{10}\implies
\begin{array}{llll}
term&coefficient&value\\
-----&-----&-----\\
1&+1&(5x)^{10}(-y)^0\\
2&+10&(5x)^9(-y)^1\\
3&+45&(5x)^8(-y)^2\\
4&+120&(5x)^7(-y)^3\\
5&+210&(5x)^6(-y)^4\\
6&+252&(5x)^5(-y)^5\\
7&+210&(5x)^4(-y)^6\\
8&+120&(5x)^3(-y)^7\\ 9&+45&(5x)^2(-y)^8
\end{array}[/tex]
now, how do we get the coefficients? well, the first coefficient is 1, any subsequent is " the product of the current terms's coefficient and the exponent of the first element, divided by the exponent of the second element in the next term", now that's a mouthful, but for example,
how did get 210 for the 5th expanded term? well is just 120 * 7 / 4
how about 252 of the 6th term? 210 * 6 / 5.
how about 45 of the 9th one? 120 * 3 / 8.
of course, the exponents for each is simple, as you'd already know from the binomial theorem.
so, just expand away the 9th expanded term.
