At a party, each person brings 1 packet of sweets and 1 chocolate bar. The packets of sweets and the chocolate bars have to be shared out among the people at the party such that each person gets one (whole) packet of sweets and one (whole) chocolate bar, but no one gets the sweets or chocolate that they brought with them. If there are 7 people at the party, how many ways are there to do this?
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This is a "double" derangement problem, involving two separate and independent derangements of 7 items each. As the number of derangements of a set of 7 items is 1 8 5 4 , the desired solution is 1 8 5 4 2 = 3 4 3 7 3 1 6 .