[tex]\bf \left( 2p-\frac{1}{2}q \right)^{10}\implies
\begin{array}{llll}
term&coefficient&value\\
-----&-----&-----\\
1&+1&(2p)^{10}(-\frac{1}{2}q)^0\\
2&+10&(2p)^9(-\frac{1}{2}q)^1\\
3&+45&(2p)^8(-\frac{1}{2}q)^2\\
4&+120&(2p)^7(-\frac{1}{2}q)^3\\
5&+210&(2p)^6(-\frac{1}{2}q)^4\\
6&+252&(2p)^5(-\frac{1}{2}q)^5
\end{array}[/tex]
how do we get the coefficient for the each term? well, the first coefficient is 1, and all subsequents are "the product of the current coefficient and the exponent of the first element divided by the exponent of the second element on the next term", so, that's a mouthful, but for example,
for the 5th term of the expansion, how did we get 210? well, is just 120 * 7 / 4.
for the 6th term, how did we get 252? well, is just 210 * 6 / 5.
recall that if the exponent is 10, that means the expansion will be 11 terms, and therefore the middle term will be the 6th one.
so, just combine the 6th term away.
