Giving Presents
couples are trading presents as follows: the couples give each other a present, then each person gives another person of the opposite gender (not including his/her significant other) a present such that every person gets presents, including the present that couples traded each other. How many ways can this be done?
The answer is 81.
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1 solution
We can disregard the couples' exchange of presents, since there is only one way for that to happen...
Looking at the females' side first:
The number of ways that the females can give presents to the males is almost a permutation, but it is not allowed for a female to give her significant other the present, and so we are actually counting derangements. The n th derangement can be calculated as ⌊ e n ! ⌉ . With 4 women, there are 9 derangements.
For the males, there are also 9 derangements.
So the answer is 9 × 9 = 8 1