Question

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is

Answer 1

553

Answer 2

To find the serial number (rank) of the word "GTWENTY" when all its letters are arranged in alphabetical order, we follow a systematic approach based on permutations.

The letters in the word "GTWENTY" are G, T, W, E, N, T, Y. Let's list them and their frequencies: G: 1 T: 2 W: 1 E: 1 N: 1 Y: 1 Total number of letters = 7. The letter 'T' is repeated twice.

First, arrange the unique letters in alphabetical order: E, G, N, T, W, Y. When considering the repeated letter 'T', the full sorted list of letters is: E, G, N, T, T, W, Y.

We will count the number of words that come before "GTWENTY" in a dictionary.

1. Words starting with a letter alphabetically smaller than 'G' (the first letter of GTWENTY): The only letter smaller than 'G' is 'E'. If the word starts with 'E', the remaining 6 letters are G, N, T, T, W, Y. The number of permutations of these 6 letters, where 'T' repeats twice, is: $\frac{6!}{2!} = \frac{720}{2} = 360$ words.

2. Words starting with 'G' (the first letter of GTWENTY): Now, fix 'G' at the first position. The remaining letters are T, W, E, N, T, Y. Arrange these remaining letters alphabetically: E, N, T, T, W, Y. The second letter of "GTWENTY" is 'T'. Count words where the second letter is alphabetically smaller than 'T':

   • Words starting with 'GE': If the word starts with 'GE', the remaining 5 letters are N, T, T, W, Y. The number of permutations of these 5 letters, where 'T' repeats twice, is: $\frac{5!}{2!} = \frac{120}{2} = 60$ words.
   • Words starting with 'GN': If the word starts with 'GN', the remaining 5 letters are E, T, T, W, Y. The number of permutations of these 5 letters, where 'T' repeats twice, is: $\frac{5!}{2!} = \frac{120}{2} = 60$ words.

3. Words starting with 'GT' (the first two letters of GTWENTY): Now, fix 'GT' at the first two positions. The remaining letters are W, E, N, T, Y. (Note: one 'T' has been used, one 'T' remains). Arrange these remaining letters alphabetically: E, N, T, W, Y. The third letter of "GTWENTY" is 'W'. Count words where the third letter is alphabetically smaller than 'W':

   • Words starting with 'GTE': If the word starts with 'GTE', the remaining 4 letters are N, T, W, Y. These 4 letters are distinct, so the number of permutations is: $4! = 24$ words.
   • Words starting with 'GTN': If the word starts with 'GTN', the remaining 4 letters are E, T, W, Y. These 4 letters are distinct, so the number of permutations is: $4! = 24$ words.
   • Words starting with 'GTT': If the word starts with 'GTT', the remaining 4 letters are E, N, W, Y. These 4 letters are distinct, so the number of permutations is: $4! = 24$ words.

4. Words starting with 'GTW' (the first three letters of GTWENTY): Now, fix 'GTW' at the first three positions. The remaining letters are E, N, T, Y. Arrange these remaining letters alphabetically: E, N, T, Y. The fourth letter of "GTWENTY" is 'E'. There are no letters smaller than 'E' in the remaining set.

5. Words starting with 'GTWE' (the first four letters of GTWENTY): Now, fix 'GTWE' at the first four positions. The remaining letters are N, T, Y. Arrange these remaining letters alphabetically: N, T, Y. The fifth letter of "GTWENTY" is 'N'. There are no letters smaller than 'N' in the remaining set.

6. Words starting with 'GTWEN' (the first five letters of GTWENTY): Now, fix 'GTWEN' at the first five positions. The remaining letters are T, Y. Arrange these remaining letters alphabetically: T, Y. The sixth letter of "GTWENTY" is 'T'. There are no letters smaller than 'T' in the remaining set.

7. Words starting with 'GTWENT' (the first six letters of GTWENTY): Now, fix 'GTWENT' at the first six positions. The remaining letter is Y. Arrange this remaining letter alphabetically: Y. The seventh letter of "GTWENTY" is 'Y'. There are no letters smaller than 'Y' in the remaining set.

We have now reached the word "GTWENTY".

Total number of words before "GTWENTY" is the sum of the counts from steps 1, 2, and 3: Total words = $360 + 60 + 60 + 24 + 24 + 24 = 552$.

The serial number of the word "GTWENTY" is one more than the number of words preceding it. Serial number = $552 + 1 = 553$.