JEE Mathematics 2024 Question with Solution

Given: The letters of the word GTWENTY are to be arranged in dictionary order. The letter T is repeated twice.

Find: The serial number of the word GTWENTY.

Arrange the letters in alphabetical order: $E, G, N, T, T, W, Y$. Now count how many admissible words come before GTWENTY.

Words starting with E come before all words starting with G.

Now fix the first letter as G. For the second position, letters smaller than T are E and N.

Words starting with GE:

Words starting with GN:

Now fix GT. Remaining letters are $E, N, T, W, Y$. For the third position, letters smaller than W are E, N, T.

Words starting with GTE:

Words starting with GTN:

Words starting with GTT:

The word GTWENTY itself contributes $1$ to its serial number.

Therefore,

So, the serial number of GTWENTY is $553$. Hence, the correct option is A.

The second approach shown in the solution first reaches $649$, but then explicitly states that this needs rechecking and finally refines the answer to $553$. The overcount happens because repeated letters must be handled carefully at each stage of dictionary-rank counting.

For this question, the consistent count extracted from the valid working is:

• before G: $\frac{6!}{2!}=360$
• before GT: $\frac{5!}{2!}+\frac{5!}{2!}=60+60$
• before GTW: $4!+4!+4!=24+24+24$
• the word itself: $1$

Thus the final rank remains

So the extracted solution supports option A.