Question

Binomial ( 5x - y ) ^ 10 terms ax ^ 2 y ^8

Answer 1

$\bf (5x-y)^{10}\implies \begin{array}{llll} term&coefficient&value\\ -----&-----&-----\\ 1&+1&(5x)^{10}(-y)^0\\ 2&+10&(5x)^9(-y)^1\\ 3&+45&(5x)^8(-y)^2\\ 4&+120&(5x)^7(-y)^3\\ 5&+210&(5x)^6(-y)^4\\ 6&+252&(5x)^5(-y)^5\\ 7&+210&(5x)^4(-y)^6\\ 8&+120&(5x)^3(-y)^7\\ 9&+45&(5x)^2(-y)^8 \end{array}$

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.