In how many ways can 5 men and 5 women be seated at a round table if two men must not sit together?
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To solve this problem, we can use the concept of permutations with restrictions.
Step 1: Seat the women
Since the women can sit anywhere without any restrictions, we can arrange them in a circle in (5-1)! = 4! = 24 ways.
Step 2: Seat the men
To ensure that no two men sit together, we can use the concept of "stars and bars".
Place the 5 women as fixed points on the table, leaving spaces between them. For instance, we can denote the initial arrangement as:
W1 _ W2 _ W3 _ W4 _ W5
Let's use bars ("|") to represent a potential position for a man to sit. Since two men cannot sit next to each other, we need to ensure that there is at least one woman between any two men. To achieve this, we need to distribute the 5 bars among the 5 spaces between the women.
We can visualize this as arranging 5 identical bars in 5 distinct spaces:
W1 _ W2 _ W3 _ W4 _ W5
With this setup, each potential arrangement of bars represents a different seating position for the men.
For example, if we have bars placed as "| |" in this arrangement, it would indicate that the first two men sit together, which is not allowed. Thus, we need to discard such arrangements.
Using the stars and bars formula, the number of possible ways to distribute 5 identical bars among 5 distinct spaces is given by:
C(n + k - 1, k) = C(5 + 5 - 1, 5) = C(9,5) = 126
So, there are 126 ways to seat the men such that no two men sit together.
Step 3: Calculate the total number of ways
Since the seating arrangement of the men and women are independent of each other, we can multiply the number of possible ways to arrange the women and men:
Total = Number of ways to arrange women x Number of ways to arrange men
= 24 x 126
= 3,024
Therefore, there are 3,024 ways to seat the 5 men and 5 women at a round table, given the condition that no two men can sit together.