Question

The number of arrangements of the letters of the word ARRANGEMENT in which two E’s do not occur adjacently is:

• A. \( \frac{9}{8} \cdot (10)! \)
• B. \( \frac{9}{4} \cdot (10)! \)
• C. \( \mathbf{\frac{9}{16} \cdot (10)!} \)
• D. \( \frac{9}{32} \cdot (10)! \)

Hint:

For problems with non-adjacent letters, subtract adjacent-case arrangements from the total.

Answer 1

The Correct Option is C

"ARRANGEMENT" has 11 letters. Frequency: A-1, R-2, N-2, G-1, E-2, M-1, T-1. Total arrangements with repeated letters: \[ \frac{11!}{2! \cdot 2! \cdot 2!} \] Now count arrangements where the two Es are adjacent: treat "EE" as one unit. We now arrange 10 units: \[ \frac{10!}{2! \cdot 2!} \] Subtract: \[ \text{Required} = \frac{11!}{2! \cdot 2! \cdot 2!} - \frac{10!}{2! \cdot 2!} \] Answer is simplified as \( \boxed{\frac{9}{16} \cdot 10!} \)