Four men and four women are to be seated alternatively at a round table. In how many ways can this be done?
Sorry no one ever answered your question bb... :(
Well, there are 8 people in all.
One way: Half of the table could have just women and the other half could have just men
Another way: 2 men and 2 women could be sitting on one half, and the other would have two women and two men.
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Where are the teachers?
tutors*
To solve this problem, we can consider the initial arrangement of the men and women as fixed, and then count the number of ways they can be seated alternatively around the circle.
Step 1: Fix one of the men (say M1) at a position on the circle. We can imagine this to be the starting position.
Step 2: Arrange the remaining three men (M2, M3, M4) in a line. There are 3! = 6 ways to do this.
Step 3: Arrange the four women (W1, W2, W3, W4) in a line. There are 4! = 24 ways to do this.
Step 4: Consider the seats between the men. There are four gaps between the men (if we imagine them seated in a line). We need to place the four women in these gaps. This can be done in C(4,4) = 1 way.
Step 5: Multiply the number of ways in each step to get the total number of arrangements. Therefore, the total number of ways is 1 * 6 * 24 * 1 = 144.
So, there are 144 ways to seat the four men and four women alternatively at a round table.