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If this is what the question means, then you need to know the number of permutations (arrangements) of items that contain items repeated of the first kind ('b') and items repeated of the second kind ('i'). This is(For a general formula see this page.)
Since just one of these arrangements is the correct one, the probability that this arrangement occurs at random is2) Again, if I may re-word the question: the names of 4 children are randomly selected by each child. What is the probability that none of the children selects their own name?
Such a selection - where no item is chosen in its 'natural' place - is called a derangement. With items, there are derangements. (See, for example, just here.)
It doesn't take long to list all the possibilities with 4 items, but if you want the formula for , the number of derangements of items, it is in the form of a recurrence relation:with defined asYou'll see that this gives:and so on.
Since the number of arrangements of items is , the probability that one chosen at random is a derangement is clearly