In how many ways can the acronym NOUN be written?

1 Answer

#(4!)/(2!)=4xx3=12#

Explanation:

If we had a series of letters that were all different, such as POUR, we could arrange them in #4! =4xx3xx2=24# ways (in that the first letter could be any of the 4, then the next letter could be any of the remaining 3, and so on).

But we have 2 Ns and they are indistinguishable from each other and so we need to take out all the arrangements that will be double counted if we don't (NOUN is the same as NOUN, even though I used the first N first in the first example and put the first N second in the second example).

We take out these duplicates by dividing by the factorial of the number of duplicates, which gives us:

#(4!)/(2!)=4xx3=12#