Question

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

Answer 1

361

Answer 2

The letters of the word "GTHIRTY" in alphabetical order are G, H, I, R, T, T, Y. There are 7 letters, with 'T' repeated twice.

To find the serial number of "GTHIRTY", we count the number of words that come before it in the dictionary.

1. Words starting with a letter smaller than 'G': There are no such letters. (Count = 0)

2. Words starting with 'G': The remaining letters are H, I, R, T, T, Y (6 letters, with 'T' repeated twice). The number of permutations is $\frac{6!}{2!} = \frac{720}{2} = 360$.

3. Words starting with 'GT': The next letter in "GTHIRTY" is 'H'. We consider letters smaller than 'H' from the remaining set {H, I, R, T, Y}. There are no such letters. (Count = 0)

4. Words starting with 'GTH': The next letter is 'I'. We consider letters smaller than 'I' from the remaining set {I, R, T, Y}. There are no such letters. (Count = 0)

5. Words starting with 'GTHI': The next letter is 'R'. We consider letters smaller than 'R' from the remaining set {R, T, Y}. There are no such letters. (Count = 0)

6. Words starting with 'GTHIR': The next letter is 'T'. We consider letters smaller than 'T' from the remaining set {T, Y}. There are no such letters. (Count = 0)

7. Words starting with 'GTHIRT': The next letter is 'Y'. We consider letters smaller than 'Y' from the remaining set {Y}. There are no such letters. (Count = 0)

Total number of words before "GTHIRTY" = $0 + 360 + 0 + 0 + 0 + 0 + 0 = 360$. The serial number of "GTHIRTY" is the total count of preceding words plus 1. Serial Number = $360 + 1 = 361$.