1. **State the problem:** Find the number of combinations of nickels (5 cents) and quarters (25 cents) that total 80 cents using exactly 10 coins. 2. **Define variables:** Let $n$ be the number of nickels and $q$ be the number of quarters. 3. **Write the system of equations:** $$n + q = 10$$ $$5n + 25q = 80$$ 4. **Solve the first equation for $n$:** $$n = 10 - q$$ 5. **Substitute $n$ into the second equation:** $$5(10 - q) + 25q = 80$$ 6. **Simplify:** $$50 - 5q + 25q = 80$$ $$50 + 20q = 80$$ 7. **Isolate $q$:** $$20q = 80 - 50$$ $$20q = 30$$ $$q = \frac{30}{20}$$ $$q = \frac{\cancel{30}}{\cancel{20}} \times \frac{3}{2} = 1.5$$ 8. **Interpretation:** $q$ must be an integer number of coins, but $1.5$ is not an integer, so no solution here. 9. **Check integer values of $q$ from 0 to 10:** - For $q=1$, $n=9$, total value $5(9)+25(1)=45+25=70$ cents - For $q=2$, $n=8$, total value $5(8)+25(2)=40+50=90$ cents - For $q=3$, $n=7$, total value $35+75=110$ cents - For $q=4$, $n=6$, total value $30+100=130$ cents - For $q=5$, $n=5$, total value $25+125=150$ cents - For $q=6$, $n=4$, total value $20+150=170$ cents - For $q=7$, $n=3$, total value $15+175=190$ cents - For $q=8$, $n=2$, total value $10+200=210$ cents - For $q=9$, $n=1$, total value $5+225=230$ cents - For $q=10$, $n=0$, total value $0+250=250$ cents None equal 80 cents. 10. **Conclusion:** There are no combinations of nickels and quarters totaling 80 cents with exactly 10 coins. **Final answer:** 0 combinations.