Advent calendar 2023
20 December
There are 6 different ways that three balls labelled 1 to 3 can be put into two boxes labelled
A and B so that no box is empty:
How many ways can five balls labelled 1 to 5 be put into four boxes labelled A to D so that
no box is empty?
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One method for putting the five balls into the four boxes is:
pick one of the balls (there are 5 ways to do this), then put the other four balls into boxes with one in each box (there are 4! ways to do this), then
put the picked ball into a box (there are 4 ways to do this). This method gives exactly two ways of obtaining each possible arrangement, so the number of
ways is \(5\times4!\times4\div2\) or 240.
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