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Mr and mrs smith are in line with ten other people in how many ways can these 12 people be arranged in this line so that mr and mrs smith are not together?
In all there are 12 ! ways. If Mr and Mrs S are treated as a single unit, then there are 11! ways of arranging the 10 other people with Mr and Mrs S together.
But these can be Mr-Mrs or Mrs-Mr so, in fact, there are 2*11! ways in which Mr and Mrs S are together.
So the number of ways in which they are NOT together is 12! - 2*11!
= 399,168,000
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