Q.Find the $5^{\text{th}}$ term in the expansion of $(x-2y)^{13}$.

The Binomial Theorem expands $(x+y)^n$, for a positive integer $n$, directly as a sum of $n+1$ terms without repeated multiplication: $(x+y)^n = \sum_{r=0}^{n} \,^{n}C_{r}\,x^{n-r}y^{r}$. Its coefficients are exactly the binomial coefficients of Pascal's triangle. The general term, $T_{r+1} = \,^{n}C_{r}\,x^{n-r}y^{r}$, is the tool used to extract any single specific term …