1. **State the problem:** Find the coefficient of $x^{12}y^{13}$ in the expansion of $(x+y)^{25}$. 2. **Formula used:** The binomial theorem states that $$ (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} $$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient. 3. **Identify terms:** We want the term where the power of $x$ is 12 and the power of $y$ is 13. Since the total power is 25, this matches $k=12$ because $x^{12}y^{25-12} = x^{12}y^{13}$. 4. **Calculate the coefficient:** $$ \binom{25}{12} = \frac{25!}{12! \times 13!} $$ 5. **Evaluate the binomial coefficient:** Using factorial values or a calculator, $$ \binom{25}{12} = 5200300 $$ 6. **Final answer:** The coefficient of $x^{12}y^{13}$ in $(x+y)^{25}$ is **5200300**.