Distinct arrangements of the letters in MISSISSIPPI

Question

Consider all arrangements of the 11 letters of the word 	ext{MISSISSIPPI}, counting arrangements that read as the same string only once. How many distinct arrangements are there?

Accepted Answer

Tally the letters: M appears 1 time, I appears 4 times, S appears 4 times, and P appears 2 times (1 + 4 + 4 + 2 = 11). Start from 11! and divide out the interchangeable copies of each repeated letter: \frac{11!}{4!\,4!\,2!\,1!} = \frac{39916800}{24 	imes 24 	imes 2} = \frac{39916800}{1152} = 34650. Each factorial in the denominator cancels the orderings among identical letters that we should not count as different. Takeaway. For several repeated letters, divide n! by the factorial of each count. MISSISSIPPI gives \dfrac{11!}{4!\,4!\,2!} = 34650.

Distinct arrangements of the letters in MISSISSIPPI

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