Final answer:
The total number of dances is calculated by adding the 25 dances between men and women to the 10 unique dances among the women, resulting in 35 total dances.
Step-by-step explanation:
To calculate the total number of dances that occurred, we need to consider two separate sets of dances: those between men and women, and those between the women themselves.
First, for the dances between men and women: Each of the 5 men danced with each of the 5 women. This can be calculated as 5 men x 5 women = 25 dances.
Next, for dances among the women: In this case, we have to find the number of combinations where each woman dances with each other woman exactly once. Since there are 5 women, we can use the combination formula nC2, which is equal to n!/(2!(n-2)!), where n is the number of items to choose from, in this case, 5.
So, 5C2 = 5!/(2!(5-2)!) = (5x4)/(2x1) = 10. Thus, there are 10 unique pairs of women dancing with each other, making 10 dances.
Adding the two sets of dances together: 25 dances (men with women) + 10 dances (women with each other) = 35 total dances.