In how many different ways can the word 'EATERANEATER' be arranged so that no two E's occur together.

Answer

To solve the problem of arranging the letters in the word "EATERANEATER" such that no two E's are together, we can follow these steps:

### Step 1: Count the letters
The word "EATERANEATER" consists of 12 letters:
- E: 4 times
- A: 3 times
- T: 2 times
- R: 2 times
- N: 1 time

### Step 2: Arrange the non-E letters
First, we will arrange the letters excluding E. The remaining letters are A, A, A, T, T, R, R, N, which totals to 8 letters.

The total arrangements of these letters can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{8!}{3! \times 2! \times 2!} \]
Where:
- \(8!\) is the factorial of the total letters,
- \(3!\) accounts for the 3 A's,
- \(2!\) accounts for the 2 T's,
- \(2!\) accounts for the 2 R's.

### Step 3: Calculate the arrangements
Calculating the values:
\[ 8! = 40320 \]
\[ 3! = 6, \quad 2! = 2 \]
Thus,
\[ \text{Total arrangements} = \frac{40320}{6 \times 2 \times 2} = \frac{40320}{24} = 1680 \]

### Step 4: Determine the gaps for E's
When we arrange the 8 letters (A, A, A, T, T, R, R, N), we create gaps where E's can be placed. There will be 9 gaps:
- Before the first letter
- Between each pair of letters (7 gaps)
- After the last letter

### Step 5: Place the E's in the gaps
We need to choose 4 gaps from the 9 available gaps to place the E's such that no two E's are together. The number of ways to choose 4 gaps from 9 is given by the combination formula:
\[ \binom{9}{4} \]

### Step 6: Calculate the combinations
Calculating \( \binom{9}{4} \):
\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

### Step 7: Total arrangements
Finally, the total arrangements of the word "EATERANEATER" such that no two E's are together is given by multiplying the arrangements of the non-E letters by the ways to place the E's:
\[ \text{Total arrangements} = 1680 \times 126 = 211680 \]

### Final Answer
Thus, the total number of different ways to arrange the word "EATERANEATER" such that no two E's occur together is **211680**.

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