1. Problem 213: Find the least possible value of the product of 20 integers chosen with repetition from consecutive integers -10 to 10 inclusive. - The integers range from -10 to 10, including zero. - We choose 20 integers with repetition allowed. - We want to minimize the product. 2. Key points: - Multiplying by zero yields zero. - Multiplying an even number of negative numbers yields a positive product. - Multiplying an odd number of negative numbers yields a negative product. - The smallest magnitude product is zero if zero is chosen. - The largest magnitude negative product is when we have an odd number of -10's and the rest 10's. 3. Consider options: - (-10)^20 = positive large number. - (-10)^10 = positive large number. - 0 = product zero. - -(10)^19 = negative large number. - -(10)^20 = negative large number but even power, so positive actually, so invalid. 4. Since we want the least possible value (most negative), the product with an odd number of -10's and the rest 10's gives the most negative large number. 5. The least possible value is thus $-(10)^{19}$. --- 6. Problem 214: Number of 5-letter strings from letters D, G, I, I, T where the two I's are separated by at least one letter. 7. Total permutations of letters D, G, I, I, T: $$\frac{5!}{2!} = \frac{120}{2} = 60$$ 8. Number of permutations where the two I's are together: - Treat II as one letter, so letters are {II, D, G, T}. - Number of permutations: $4! = 24$ 9. Number of permutations where I's are separated by at least one letter: $$60 - 24 = 36$$ --- 10. Problem 215: Calculate $$\frac{0.99999999}{1.0001} - \frac{0.99999991}{1.0003}$$ 11. Approximate each term using binomial expansion for small $x$: - $\frac{0.99999999}{1.0001} \approx 0.99999999 \times (1 - 0.0001) = 0.99999999 - 0.000099999999 = 0.99989999$ - $\frac{0.99999991}{1.0003} \approx 0.99999991 \times (1 - 0.0003) = 0.99999991 - 0.000299999973 = 0.99969991$ 12. Subtract: $$0.99989999 - 0.99969991 = 0.00020008 \approx 2 \times 10^{-4}$$ 13. The closest option is D: $2(10^{-4})$. Final answers: - 213: $-(10)^{19}$ - 214: 36 - 215: $2 \times 10^{-4}$