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.