answer:1. **Define the problem and use complementary counting**: We need to find the probability that each delegate sits next to at least one delegate from another country. We will use complementary counting to find the number of ways in which at least one delegate is not next to a delegate from another country. 2. **Count the total number of arrangements**: The total number of ways to arrange 9 people in a circle is given by ((9-1)!) because rotations of the same arrangement are considered identical. Therefore, the total number of arrangements is: [ 8! = 40320 ] 3. **Count the number of invalid arrangements**: An invalid arrangement is one where at least one "three-block" of delegates from the same country exists. We will use the principle of inclusion-exclusion (PIE) to count these invalid arrangements. 4. **Count arrangements with at least one "three-block"**: - Choose one country to form a "three-block". There are (binom{3}{1} = 3) ways to choose the country. - Arrange the remaining 6 delegates. There are (6!) ways to arrange them. - Arrange the "three-block" and the remaining 6 delegates. There are (3!) ways to arrange the "three-block". Therefore, the number of ways to have at least one "three-block" is: [ 3 times 3! times 6! = 3 times 6 times 720 = 12960 ] 5. **Count arrangements with at least two "three-blocks"**: - Choose two countries to form "three-blocks". There are (binom{3}{2} = 3) ways to choose the countries. - Arrange the remaining 3 delegates. There are (3!) ways to arrange them. - Arrange the "three-blocks" and the remaining 3 delegates. There are (3! times 3! times 4) ways to arrange the "three-blocks" and the remaining 3 delegates. Therefore, the number of ways to have at least two "three-blocks" is: [ 3 times 3! times 3! times 4 = 3 times 6 times 6 times 4 = 432 ] 6. **Count arrangements with all three "three-blocks"**: - Arrange the three "three-blocks" in a circle. There are (2!) ways to arrange them. - Arrange the delegates within each "three-block". There are ((3!)^3) ways to arrange them. Therefore, the number of ways to have all three "three-blocks" is: [ 2! times (3!)^3 = 2 times 6^3 = 2 times 216 = 432 ] 7. **Apply the principle of inclusion-exclusion**: Using PIE, the number of invalid arrangements is: [ 12960 - 432 + 432 = 12960 ] 8. **Calculate the number of valid arrangements**: The number of valid arrangements is the total number of arrangements minus the number of invalid arrangements: [ 40320 - 12960 = 27360 ] 9. **Calculate the probability**: The probability that each delegate sits next to at least one delegate from another country is: [ frac{27360}{40320} = frac{27360 div 720}{40320 div 720} = frac{38}{56} = frac{19}{28} ] 10. **Find (m+n)**: Since the probability is (frac{19}{28}), we have (m = 19) and (n = 28). Therefore, (m+n = 19 + 28 = 47). The final answer is ( boxed{47} )

answer:First, let's find out the length of each piece when Vincent cuts the 120 inches of rope into 12 equal pieces. 120 inches / 12 pieces = 10 inches per piece Now, Vincent ties four pieces together. Before tying, the total length of the four pieces would be: 4 pieces * 10 inches per piece = 40 inches However, each knot causes a 10% loss in length. Since he is tying four pieces together, there will be three knots (since the first piece is tied to the second, the second to the third, and the third to the fourth). Each knot causes a loss of 10% of the length of a piece, which is: 10% of 10 inches = 0.10 * 10 inches = 1 inch lost per knot Since there are three knots, the total length lost due to knots is: 3 knots * 1 inch per knot = 3 inches Now, subtract the total length lost from the original total length of the four pieces: 40 inches - 3 inches = 37 inches So, after tying four pieces together and accounting for the loss due to knots, the length of the tied-together pieces is boxed{37} inches.

answer:To find the quotient when the number 127 is divided by 25, we can use the following formula: Quotient = (Original number - Remainder) / Divisor Given that the original number is 127, the remainder is 2, and the divisor is 25, we can plug these values into the formula: Quotient = (127 - 2) / 25 Quotient = 125 / 25 Quotient = 5 Therefore, the quotient is boxed{5} .

answer:Let's start with the expressions (PQ) and (RS). 1. (P cdot Q = (sqrt{2023} + sqrt{2022})(sqrt{2023} - sqrt{2022})) Using the difference of squares formula, (a^2 - b^2): [ PQ = (2023 - 2022) = 1 ] 2. (R cdot S = (sqrt{2023} + sqrt{2024})(sqrt{2023} - sqrt{2024})) Again, applying the difference of squares: [ RS = (2023 - 2024) = -1 ] 3. Now, find the product ((P cdot Q) cdot (R cdot S)): [ (P cdot Q) cdot (R cdot S) = 1 cdot (-1) = -1 ] So the answer is (boxed{-1}). Conclusion: This solution shows (PQ = 1) and (RS = -1), which are consistent with the property of the difference of squares when applied to square root expressions of consecutive integers. The final answer, (-1), is derived correctly from these results.