Math  /  Discrete

QuestionHow many ways can the letters in MAXIMUM be arranged? Well, there are 7 letters, so that would be 7!7! if they were all different. But that counts MAXIMUM as a new arrangement (switching the M's around). To avoid recounting these "arrangements," note that every ordering appears 3!=63!=6 times because of the matching M's. To eliminate all those duplicates from consideration, divide by 3!. This reveals that there are actually 840 ( 7 ! divided by 3 !) distinct permutations of the letters in MAXIMUM. Similarly, there would be 1260 ( 7!7! divided by 212!) different ways to arrange the letters in DOTPLOT. There would be 7!7! arrangements if they were all different, reduced by a factor of 2!2! for the matching O's and 2!2! for the matching Ts. Complete parts a through f below. a) How many ways can the letters in the word OHIO be arranged? \square

Studdy Solution
Divide the total permutations by the factorial of the repetitions:
242=12 \frac{24}{2} = 12
The number of distinct ways to arrange the letters in the word "OHIO" is:
12 \boxed{12}

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