The letters of the word 'RANDOM' are written in all possible orders and these words are written out as in dictionary. Find the rank of the word 'RANDOM'.

Answer

To find the rank of the word "RANDOM" when all its letters are arranged in alphabetical order, we can follow these steps:

### Step 1: Identify the letters and their order
The letters in "RANDOM" are A, D, M, N, O, R. Arranging them in alphabetical order gives us: A, D, M, N, O, R.

### Step 2: Count the arrangements starting with letters before 'R'
1. **Words starting with 'A':**
- Remaining letters: D, M, N, O, R
- Number of arrangements = 5! = 120

2. **Words starting with 'D':**
- Remaining letters: A, M, N, O, R
- Number of arrangements = 5! = 120

3. **Words starting with 'M':**
- Remaining letters: A, D, N, O, R
- Number of arrangements = 5! = 120

4. **Words starting with 'N':**
- Remaining letters: A, D, M, O, R
- Number of arrangements = 5! = 120

5. **Words starting with 'O':**
- Remaining letters: A, D, M, N, R
- Number of arrangements = 5! = 120

### Step 3: Count the arrangements starting with 'R'
Now we focus on words starting with 'R'.

1. **Fix 'R' and consider the next letter:**
- The next letter must be 'A' (the only letter before D, M, N, O):
- Remaining letters: D, M, N, O

2. **Fix 'RA' and consider the next letter:**
- The next letter can be 'D':
- Remaining letters: M, N, O
- Number of arrangements = 3! = 6

3. **Fix 'RAD' and consider the next letter:**
- The next letter can be 'M':
- Remaining letters: N, O
- Number of arrangements = 2! = 2

4. **Fix 'RAM' and consider the next letter:**
- The next letter can be 'N':
- Remaining letters: O
- Number of arrangements = 1! = 1

5. **Fix 'RAN' and consider the next letter:**
- The next letter must be 'D':
- Remaining letters: O, M
- Number of arrangements = 2! = 2

6. **Fix 'RAND' and consider the next letter:**
- The next letter must be 'O':
- Remaining letter: M
- Number of arrangements = 1! = 1

### Step 4: Total the arrangements
Now we can sum all the arrangements:
- Words starting with 'A': 120
- Words starting with 'D': 120
- Words starting with 'M': 120
- Words starting with 'N': 120
- Words starting with 'O': 120
- Words starting with 'RA': 6
- Words starting with 'RAD': 2
- Words starting with 'RAM': 1
- Words starting with 'RAN': 2
- Words starting with 'RAND': 1

Total = 120 + 120 + 120 + 120 + 120 + 6 + 2 + 1 + 2 + 1 = 614

### Conclusion
The rank of the word "RANDOM" is **614**.

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