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

Answer

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

### Step-by-Step Solution:

1. **List the letters in alphabetical order**: The letters a, b, c, d, e are already in alphabetical order.

2. **Count permutations starting with letters before 'd'**:
- If we fix 'a' in the first position, the remaining letters are b, c, d, e. The number of permutations of these 4 letters is \(4! = 24\).
- If we fix 'b' in the first position, the remaining letters are a, c, d, e. The number of permutations is also \(4! = 24\).
- If we fix 'c' in the first position, the remaining letters are a, b, d, e. The number of permutations is again \(4! = 24\).

Total permutations before 'd' = \(24 + 24 + 24 = 72\).

3. **Fix 'd' in the first position**: Now we focus on permutations starting with 'd'.

4. **Count permutations starting with 'da'**:
- If we fix 'a' in the second position, the remaining letters are b, c, e. The number of permutations is \(3! = 6\).

5. **Count permutations starting with 'db'**:
- If we fix 'b' in the second position, the remaining letters are a, c, e. The number of permutations is \(3! = 6\).

6. **Count permutations starting with 'dc'**:
- If we fix 'c' in the second position, the remaining letters are a, b, e. The number of permutations is \(3! = 6\).

7. **Fix 'd' and 'e' in the first two positions**: Now we have 'de' fixed.

8. **Count permutations starting with 'deb'**:
- If we fix 'b' in the third position, the remaining letters are a, c. The number of permutations is \(2! = 2\).

9. **Count permutations starting with 'dec'**:
- If we fix 'c' in the third position, the remaining letters are a, b. The number of permutations is \(2! = 2\).

10. **Fix 'deb' and count permutations**:
- Now, we have 'deb' fixed. The next letter must be 'a' (since we want 'debac'), and the only letter left is 'c'.
- The arrangement 'debac' is the only permutation left, which counts as \(1! = 1\).

11. **Calculate the total rank**:
- Total permutations before 'd': \(72\)
- Total permutations starting with 'da': \(6\)
- Total permutations starting with 'db': \(6\)
- Total permutations starting with 'dc': \(6\)
- Total permutations starting with 'de': \(2\) (for 'deb' and 'dec')
- Total permutations for 'deb': \(1\)

Therefore, the rank of 'debac' is:
\[ 72 + 6 + 6 + 6 + 2 + 1 = 93 \]

### Final Answer:
The rank of the permutation "debac" is **93**.

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