If all word by using letters of word "GTWENTY" are arranged as in dictionary, then rank of the word "GTWENTY" is

Options

A. 552
B. 553
C. 554
D. 551

Answer

To find the rank of the word "GTWENTY" when all permutations of its letters are arranged in dictionary order, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the letters in "GTWENTY":**
The letters are G, T, W, E, N, T, Y.

2. **Sort the letters alphabetically:**
The sorted order of the letters is: E, G, N, T, T, W, Y.

3. **Count the total permutations starting with letters before 'G':**
- **Starting with 'E':**
- Remaining letters: G, N, T, T, W, Y (6 letters, with T repeating twice).
- The number of permutations = \( \frac{6!}{2!} = \frac{720}{2} = 360 \).

4. **Count the permutations starting with 'G':**
- **Next letter after 'G':**
- The next letter in "GTWENTY" is 'T'. We need to count permutations starting with 'G' and letters before 'T'.
- **Starting with 'G' and 'E':**
- Remaining letters: N, T, T, W, Y (5 letters, with T repeating twice).
- The number of permutations = \( \frac{5!}{2!} = \frac{120}{2} = 60 \).

5. **Count permutations starting with 'G' and 'N':**
- **Starting with 'G' and 'N':**
- Remaining letters: T, T, W, Y (4 letters, with T repeating twice).
- The number of permutations = \( \frac{4!}{2!} = \frac{24}{2} = 12 \).

6. **Count permutations starting with 'G' and 'T':**
- **Next letter after 'G' and 'T':**
- The next letter in "GTWENTY" is 'W'. We need to count permutations starting with 'G', 'T', and letters before 'W'.
- **Starting with 'G', 'T', and 'E':**
- Remaining letters: N, T, W, Y (4 letters, with T repeating twice).
- The number of permutations = \( 3! = 6 \).

7. **Count permutations starting with 'G', 'T', and 'N':**
- **Starting with 'G', 'T', and 'N':**
- Remaining letters: T, W, Y (3 letters, with T repeating twice).
- The number of permutations = \( 3! = 6 \).

8. **Count permutations starting with 'G', 'T', and 'T':**
- **Next letter after 'G', 'T', and 'T':**
- The next letter in "GTWENTY" is 'W'.
- **Starting with 'G', 'T', 'T', and 'E':**
- Remaining letters: W, Y (2 letters).
- The number of permutations = \( 2! = 2 \).

9. **Finally, we have reached "GTWENTY":**
- Now we can count the total permutations before "GTWENTY":
- From 'E': 360
- From 'G' and 'E': 60
- From 'G' and 'N': 12
- From 'G', 'T', and 'E': 6
- From 'G', 'T', and 'N': 6
- From 'G', 'T', 'T', and 'E': 2

10. **Calculate the total rank:**
- Total permutations before "GTWENTY" = 360 + 60 + 12 + 6 + 6 + 2 = 446.
- Therefore, the rank of "GTWENTY" = 446 + 1 = 447.

### Final Answer:
The rank of the word "GTWENTY" is **447**.