1 pointIn a restaurant, 10 men and 6 women are seated on 16 chairs at a round table. Find the total number of possible ways such that 10 men are always sitting next to each other.
Question
1 pointIn a restaurant, 10 men and 6 women are seated on 16 chairs at a round table. Find the total number of possible ways such that 10 men are always sitting next to each other.
Solution
To solve this problem, we can consider the 10 men as a single entity. So, we have 7 entities in total (6 women and 1 group of men) to arrange around the round table.
Step 1: Arranging 7 entities around a round table The number of ways to arrange n entities around a round table is (n-1)!. So, the number of ways to arrange 7 entities is (7-1)! = 6! = 720.
Step 2: Arranging 10 men within their group The number of ways to arrange 10 men is 10! = 3,628,800.
Step 3: Multiplying the results The total number of ways to arrange 10 men and 6 women such that all men are always sitting next to each other is the product of the results from step 1 and step 2. So, the total number of ways is 720 * 3,628,800 = 2,612,736,000.
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