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1
Two individuals are seated at a circular table with 16 people. What are the odds against them sitting together?
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Solution: Step 1: The total number of ways 16 people can be seated around a circular table is (16-1)! = 15!. Step 2: If two individuals sit together, we consider them as one entity, so there are 15 entities to be seated. Step 3: The number of ways they can be seated together is 2! * 14!. Step 4: The probability of them sitting together is (2! * 14!) / 15! = 2 / 15. Step 5: The odds against an event are given by the ratio of unfavorable outcomes to favorable outcomes. Step 6: Odds against them sitting together = (15 - 2) / 2 = 13 / 2. Step 7: Therefore, the odds against them sitting together are 13:2.
2
Bhaskar and Shakuntala are seated at a circular table with 16 people. What are the odds against Bhaskar and Shakuntala sitting together?
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Solution: Step 1: Calculate the total number of ways 16 people can be seated around a circular table: (16-1)! = 15!. Step 2: Consider Bhaskar and Shakuntala as a single entity, so there are 15 entities to be seated. Step 3: The number of ways Bhaskar and Shakuntala can be seated together is 2 * 14!. Step 4: The probability of Bhaskar and Shakuntala sitting together is (2 * 14!) / 15! = 2 / 15. Step 5: The odds against Bhaskar and Shakuntala sitting together are (15 - 2) / 2 = 13 / 2. Step 6: Therefore, the odds against Bhaskar and Shakuntala sitting together are 13:2.
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