Final answer:
To arrange 6 men and 4 women in a row so no two women sit together, arrange the men in 6! ways, then place the women in the 4 of the 7 possible gaps.
There are a total of 6! × C(7, 4) = 25,200 different ways to arrange them.
Step-by-step explanation:
The question involves finding the number of ways to arrange 6 men and 4 women in a row so that no two women sit together. This is a combinatorics problem. To solve it, we first arrange the men and then place the women in the gaps between them.
To start, we can arrange the 6 men in 6! (6 factorial) ways. Since there are 6 men, there are 7 places where the women can be seated (one on each end and between each pair of men).
However, since no two women can sit together, we can only choose 4 of these 7 places for the women.
The number of ways to choose these 4 places is given by the combination formula C(7, 4), which is the number of ways to choose 4 places out of 7 without regard to order.
Finally, we find the product of these two quantities to find the total number of arrangements where no two women sit together: 6! × C(7, 4).
Therefore, the total number of ways they can be seated is 720 × 35, which equals 25,200.