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How many visually different arrangements can be made from the letters of the word giggling?
Valget
Answer: 840. If all the letters in "giggling" were different, there would be 8*7*6*5*4*3*2*1=40320 arrangements: 8 ways to choose the first letter in an arrangement, 7 ways to choose the second, 6 ways to choose the third, and so on. But there are two problems with that: the g's and the i's. Let's start with the i's. There are two of them, which can be switched in any arrangement without changing how it looks. So when we counted 40320 arrangements above, we counted each arrangement twice, once for each order of the i's. So we should divide 40320 by 2, to get 20160. The g's are a similar idea, except there are 4 of them. So there are 4*3*2*1 or 24 visually identical ways to order the g's. That means we counted each truly different arrangement 24 times: one for each such visually identical order of the g's. So the number 20160 is inflated by a factor of 24, and we should divide it by 24, getting 840.
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