Numerical answer
In how many distinct ways can the letters of the word COMMITTEE be arranged so that the two M's are not next to each other?
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Answer: 35280
COMMITTEE has 9 letters: C, O, M, M, I, T, T, E, E (M, T, E each repeated twice).
Total arrangements = 9!/(2!×2!×2!) = 362880/8 = 45,360.
Arrangements with the two M's together: treat MM as one block, leaving 8 units (MM, C, O, I, T, T, E, E) with T and E still repeated → 8!/(2!×2!) = 40320/4 = 10,080.
Required = 45,360 − 10,080 = 35,280.
Total arrangements = 9!/(2!×2!×2!) = 362880/8 = 45,360.
Arrangements with the two M's together: treat MM as one block, leaving 8 units (MM, C, O, I, T, T, E, E) with T and E still repeated → 8!/(2!×2!) = 40320/4 = 10,080.
Required = 45,360 − 10,080 = 35,280.
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