1. **Problem:** In a 10-item Mathematics problem-solving test, how many ways can you select 5 problems to solve? 2. **Formula:** The number of ways to choose $k$ items from $n$ items without regard to order is given by the combination formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ 3. **Apply the formula:** Here, $n=10$ and $k=5$. $$\binom{10}{5} = \frac{10!}{5!5!}$$ 4. **Calculate factorials:** $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5!$ So, $$\binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5! \times 5!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1}$$ 5. **Simplify numerator and denominator:** Calculate numerator: $10 \times 9 = 90$, $90 \times 8 = 720$, $720 \times 7 = 5040$, $5040 \times 6 = 30240$ Calculate denominator: $5 \times 4 = 20$, $20 \times 3 = 60$, $60 \times 2 = 120$, $120 \times 1 = 120$ 6. **Divide:** $$\frac{30240}{120} = 252$$ **Final answer:** There are 252 ways to select 5 problems from 10.