If the permutations of a, b, c, d, e taken all together be written dowan in alphabetical order as in dictionary and numbered then the rank of the permutation debac, is

Options

A. 90
B. 91
C. 92
D. 93

Explanation

To find the rank of the permutation "debac" when all permutations of the letters a, b, c, d, and e are arranged in alphabetical order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters a, b, c, d, and e in alphabetical order are:
- a, b, c, d, e

### Step 2: Count permutations starting with letters before 'd'
1. **Starting with 'a':**
- The remaining letters are b, c, d, e.
- The number of permutations = 4! = 24.

2. **Starting with 'b':**
- The remaining letters are a, c, d, e.
- The number of permutations = 4! = 24.

3. **Starting with 'c':**
- The remaining letters are a, b, d, e.
- The number of permutations = 4! = 24.

### Step 3: Count permutations starting with 'd' and the next letter before 'e'
Now we consider permutations starting with 'd'. The next letter in "debac" is 'e', so we need to count permutations starting with 'da', 'db', and 'dc'.

1. **Starting with 'da':**
- The remaining letters are b, c, e.
- The number of permutations = 3! = 6.

2. **Starting with 'db':**
- The remaining letters are a, c, e.
- The number of permutations = 3! = 6.

3. **Starting with 'dc':**
- The remaining letters are a, b, e.
- The number of permutations = 3! = 6.

### Step 4: Count permutations starting with 'de'
Now we consider permutations starting with 'de'. The next letter in "debac" is 'b'.

1. **Starting with 'dea':**
- The remaining letters are b, c.
- The number of permutations = 2! = 2.

### Step 5: Count permutations starting with 'deb'
Now we consider permutations starting with 'deb'. The next letter in "debac" is 'a'.

1. **Starting with 'deba':**
- The remaining letter is c.
- The number of permutations = 1! = 1.

### Step 6: Count the rank
Now we can sum all the permutations counted:

- Starting with 'a': 24
- Starting with 'b': 24
- Starting with 'c': 24
- Starting with 'da': 6
- Starting with 'db': 6
- Starting with 'dc': 6
- Starting with 'dea': 2
- Starting with 'deba': 1

Total = 24 + 24 + 24 + 6 + 6 + 6 + 2 + 1 = 93

### Conclusion
Thus, the rank of the permutation "debac" is **93**.

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