1. Arrange the letters in dictionary order

The word is GTWENTY.

Its letters are: $G, T, W, E, N, T, Y$

In alphabetical order, these are: $E < G < N < T < T < W < Y$

We need the rank (serial number) of the arrangement: $GTWENTY$ among all distinct permutations of these letters, arranged lexicographically.

Since there are two identical $T$'s, we must divide by $2!$ whenever all remaining letters include both $T$'s.

2. Count words before $GTWENTY$ letter by letter

We count how many valid distinct arrangements come before it.

Position 1: $G$

Available letters: $E, G, N, T, T, W, Y$

Letters smaller than $G$ are only $E$.

If first letter is $E$, remaining letters are $G,N,T,T,W,Y$. Number of arrangements: $\frac{6!}{2!}=\frac{720}{2}=360$

So words before ours from position 1: $360$

Position 2: $T$

Now first letter fixed as $G$. Remaining letters: $E,N,T,T,W,Y$

We need letters smaller than $T$: $E, N$.

• If second letter is $E$, remaining letters: $N,T,T,W,Y$ $\frac{5!}{2!}=60$
• If second letter is $N$, remaining letters: $E,T,T,W,Y$ $\frac{5!}{2!}=60$

Total before ours from position 2: $60+60=120$

Running total: $360+120=480$

Position 3: $W$

Prefix fixed: $GT$ Remaining letters: $E,N,T,W,Y$

Letters smaller than $W$: $E,N,T$.

Each choice leaves 4 distinct letters, so each contributes: $4!=24$

Thus total contribution: $3\times 24=72$

Running total: $480+72=552$

Position 4: $E$

Prefix fixed: $GTW$ Remaining letters: $E,N,T,Y$

There is no letter smaller than $E$.

Contribution: $0$

Position 5: $N$

Prefix fixed: $GTWE$ Remaining letters: $N,T,Y$

No letter smaller than $N$ among remaining.

Contribution: $0$

Position 6: $T$

Prefix fixed: $GTWEN$ Remaining letters: $T,Y$

No letter smaller than $T$ among remaining.

Contribution: $0$

Position 7: $Y$

Only one letter remains, so contribution is $0$.

3. Find the serial number

Number of words before $GTWENTY$ is: $552$

Therefore its rank is: $552+1=553$

4. Comparison with stored correct answer

Stored correct answer = $553$

Our derived answer is also: $553$

So the stored answer is correct.