Distinct arrangements of the letters in BALLOON

Question

Consider all arrangements of the 7 letters of the word 	ext{BALLOON}, where arrangements reading as the same string count once (the two L's are interchangeable, and so are the two O's). How many distinct arrangements are there?

Accepted Answer

BALLOON has 7 letters: B, A, two L's, two O's, N. With all letters treated as different there would be 7! = 5040 orderings, but each real arrangement is overcounted by 2! for the L's and 2! for the O's. \frac{7!}{2!\,2!} = \frac{5040}{2 	imes 2} = \frac{5040}{4} = 1260. Takeaway. Divide n! by the factorial of each repeated letter's count. BALLOON has two doubles, so \dfrac{7!}{2!\,2!} = 1260.

Distinct arrangements of the letters in BALLOON

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