Multiple choice

The total number of ways of arranging the letters $AAABBBCCDEF$ in a row such that letters $C$ are separated from one another is

  1. $277200$
  2. $138600$
  3. $453600$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total letters: 11 (3A, 3B, 2C, 1D, 1E, 1F). Arrange non-C letters: 9! / (3!3!) = 10080. There are 10 gaps to place 2 C's: 10C2 = 45. Total = 10080 * 45 = 453600.

AI explanation

To separate the two C letters, first arrange the remaining 9 letters (AAABBBDEF) in 9! / (3! 3!) equals 10080 ways. This arrangement creates 10 available gaps where the C letters can be placed so they are not adjacent. The number of ways to choose 2 gaps out of 10 is 10C2 equals 45, making the total arrangements 10080 times 45 equals 453600.

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