All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and t

Q: Wait, why do we divide by $2!$ in the calculation?

Great question! When we have repeated letters, like the two 'T's in 'GTWENTY', swapping them doesn't create a new, unique word. The formula $\frac{n!}{p!}$ accounts for this by removing the redundant permutations caused by the identical letters. If we didn't divide, we would be overcounting!

Q: Why do we stop at the word 'GTWENTY' itself and add 1?

The dictionary rank is the position of the word in the ordered list. When we calculate the number of words *before* our target, we are essentially counting all the 'predecessors'. Since our target word is the very next one in the sequence, we add 1 to include the word itself in the final rank.

Q: What if the target word had more than one set of repeated letters?

The logic remains the same! If you had, say, three 'T's and two 'E's, you would divide by $3! \times 2!$. The fundamental principle is to always divide by the factorial of the frequency of each repeated letter. It's all about ensuring every unique arrangement is counted exactly once.

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

Sigma Percentile · Accepted Answer

### The Art of Lexicographical Counting: A Journey Through 'GTWENTY'

Welcome, fellow traveler of the JEE journey! Today, we are not just solving a problem; we are stepping into the shoes of a librarian organizing a very specific, very mathematical library.

Our task is to find the dictionary rank of the word 'GTWENTY'. This is a classic combinatorics problem that tests your ability to be systematic, patient, and precise. Let us break this down into a beautiful, logical sequence.

### Phase 1: The Alphabetical Foundation

Before we can count, we must organize. Imagine all the letters of 'GTWENTY' scattered on your desk: G, T, W, E, N, T, Y.

To build a dictionary, we need order. Let us sort them alphabetically: E, G, N, T, T, W, Y.

Notice something crucial? The letter 'T' appears twice. This is the 'ghost in the machine'—the repetition that we must account for in every single calculation. If we ignore it, our count will be wrong.

### Phase 2: The Systematic Pruning

We want to find the rank of 'GTWENTY'. This means we need to count every single word that comes before it by fixing letters one by one, starting from the first position.

First, consider words starting with 'E'. If we fix 'E' in the first position, we have 6 slots remaining for the letters G, N, T, T, W, Y.

The number of such words is:

$$
\frac{6!}{2!} = 360
$$

These 360 words all come before any word starting with 'G'.

### Phase 3: The 'G' Branch

Now, we move to the first letter of our target word: 'G'. We fix 'G' in the first position and look at the second position. The dictionary will list words starting with 'GE', then 'GN', then 'GT'.

We need to count all words starting with 'GE' and 'GN'. For 'GE', we fix 'G' and 'E'. We have 5 letters left: N, T, T, W, Y.

The number of words is:

$$
\frac{5!}{2!} = 60
$$

Similarly, for 'GN', we fix 'G' and 'N'. We have 5 letters left: E, T, T, W, Y. Again, the number of words is:

$$
\frac{5!}{2!} = 60
$$

So, we have $60 + 60 = 120$ words starting with 'GE' or 'GN'.

### Phase 4: The 'GT' Branch

Now we reach the 'GT' block. Our target word is 'GTWENTY', so we fix 'G' and 'T'. We look at the third position, where the dictionary will list 'GTE', 'GTN', 'GTT', and finally 'GTW'.

We must count all words starting with 'GTE', 'GTN', and 'GTT'. For 'GTE', we fix 'G', 'T', 'E'. We have 4 letters left: N, T, W, Y. These are all distinct!

The number of words is $4! = 24$. For 'GTN', we fix 'G', 'T', 'N', leaving 4 letters: E, T, W, Y. Again, $4! = 24$.

For 'GTT', we fix 'G', 'T', 'T', leaving 4 letters: E, N, W, Y. Again, $4! = 24$. Summing these up, we have $24 + 24 + 24 = 72$ words.

### Phase 5: The Final Step

We are almost there! We have accounted for all words starting with 'E', 'GE', 'GN', 'GTE', 'GTN', and 'GTT'.

The very next word in the dictionary is 'GTWENTY' itself. This contributes exactly 1 to our rank.

Now, let us add it all up:

$$
360 + 60 + 60 + 24 + 24 + 24 + 1 = 553
$$

And there you have it! The **rank of 'GTWENTY' is 553**. This process is the essence of combinatorics: breaking a complex problem into smaller, manageable, and logical steps.

Animated Solution for Mathematics - Permutations and Combinations: 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

Enter Numerical Value:

Visualized Solution

Analyze the Word $GT W E N T Y$

Words starting with $E$

Words starting with $GE$

Words starting with $GN$

Fixing $GT$ and checking $GT E$

Checking $GT N$ and $GT T$

Reaching $GT W E N T Y$

Final Rank Calculation

The Sigma Insight: Linear Permutations

The Art of Lexicographical Counting

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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