Final answer:
To guess who the 3 married couples are in a building with 6 men and 4 women, calculate the number of combinations of 3 men (C(6,3)) and multiply it with the number of permutations of the 3 women to match them (3!), resulting in 120 ways.
Step-by-step explanation:
To determine the number of ways to guess who the married couples are among 6 men and 4 women, where there are exactly 3 married couples, we can treat this as a combinatorics problem. Since we want to form pairs (couples) from the men and women, we first select 3 men from the 6 to be part of the couples, which can be done in C(6,3) ways.
Then, we match these 3 men with the 3 women to form couples. There are 3! (3 factorial) ways to rearrange 3 objects (the women in this case), which represents the different possible pairings for the men we have chosen.
The total number of ways to guess the married couples is the product of the number of ways to choose the men and the number of ways to pair them with the women: C(6,3) * 3!. Performing the calculations, we find C(6,3) = 20 (which is calculated as 6!/(3!*(6-3)!)) and 3! = 6, so the total number of ways is 20 * 6 = 120 ways.