Finding a specific term in an expansion $(a+b)^n$ without expanding How can I find a term within an expansion without actually expanding or using Pascal's Triangle? 
For example: 5th term of 
$$
\left(\dfrac{x}{y}-\dfrac{y}{x}\right)^8
$$

$$(a+b)^n=\binom{n}{0}a^nb^0 +\binom{n}{1}a^{n-1}b^1+\binom{n}{2}a^{n-2}b^2+\binom{n}{3}a^{n-3}b^3+....\\so\\5th\\term\\is\\\binom{n}{4}a^{n-4}b^4\\(\frac{x}{y}-\frac{y}{x})^8\\5th-term\\
 \binom{8}{4}(\frac{x}{y})^{8-4}(-\frac{y}{x})^4=\\\frac{8*7*6*5}{4*3*2*1}(\frac{x}{y})^{4}(\frac{y}{x})^4=\\\frac{8*7*6*5}{4*3*2*1} $$
