If the letters of the word SACHIN are arranged in all possible ways and these words are written out

Q: Why do we use $5!$ instead of $6!$ when calculating the words starting with A?

Great question! We use $5!$ because we have fixed the letter 'A' in the first slot. Since the word has $6$ letters total, fixing one leaves $5$ positions to be filled by the remaining $5$ distinct letters. The number of ways to arrange these $5$ letters is $5 \times 4 \times 3 \times 2 \times 1 = 120$.

Q: Why do we add $1$ at the very end of the calculation?

Think of the rank as a position in a queue. We calculated that there are $600$ words that come *before* any word starting with 'S'. The very next word in the dictionary is the first word that starts with 'S', which happens to be 'SACHIN' itself. Therefore, we add $1$ to include the word itself in the count.

Q: What if the word had repeating letters, like 'BANANA'?

If there were repeating letters, the simple $n!$ formula would overcount. You would need to divide by the factorial of the frequency of each repeating letter. For example, if a letter appears $k$ times, you divide by $k!$. Always remember to account for these repetitions to get the correct rank!

Question

If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Options:

(A) 601

(B) 600

(C) 603

(D) 602

Sigma Percentile · Accepted Answer

### The Lexicographical Journey: Unlocking the Rank of SACHIN

Have you ever wondered how a dictionary knows exactly where to place a word? It is not magic; it is the beautiful, rigid logic of lexicographical order.

Today, we are going to embark on a mathematical journey to find the rank of the word **SACHIN**. Imagine you are a librarian in a universe where the only words that exist are those formed by the letters S, A, C, H, I, and N.

Your task is to find the serial number of **SACHIN** if all possible permutations were listed in a dictionary.

### Phase 1: The Alphabetical Foundation

Before we can count, we must organize. A dictionary is nothing without its alphabetical backbone.

Our letters are S, A, C, H, I, and N. Let us sort them with precision:

**A, C, H, I, N, S**

This sorted list is our map. Every word in our dictionary will follow this sequence. If we want to find the rank of **SACHIN**, we need to systematically count every word that appears before it.

### Phase 2: The Strategy of Exclusion

Our target word starts with **S**. This is a massive advantage! It means we can ignore any word that starts with **A, C, H, I,** or **N**.

We just need to calculate how many words start with each of these letters and sum them up. Let us look at the words starting with **A**.

If we fix **A** in the first slot, we have $5$ empty slots remaining. We have $5$ letters left to fill them: **C, H, I, N, S**.

The number of ways to arrange these $5$ letters is $5!$ (five factorial). Calculating this, we get:

$$5! = 5 	imes 4 	imes 3 	imes 2 	imes 1 = 120$$

So, there are exactly $120$ words that start with **A**.

Now, apply this same logic to **C, H, I,** and **N**. Each of these, when fixed in the first position, generates another $120$ words.

Since there are $5$ such letters before **S** (namely **A, C, H, I, N**), the total number of words we can skip is:

$$5 	imes 120 = 600$$

### Phase 3: The Final Arrival

We have successfully skipped $600$ words. The dictionary now turns the page to the section where every word starts with **S**.

The very first word in this section will be the one where the remaining letters are arranged in the smallest possible alphabetical order. Our remaining letters are **A, C, H, I, N**.

Arranged alphabetically, they are **A-C-H-I-N**. When we place these after our fixed **S**, we get **S-A-C-H-I-N**.

This is our target word! Because it is the very first word in the 'S' section, its position is simply the total number of words before it plus one.

$$Rank = 600 + 1 = 601$$

### The Takeaway

Mathematics is often about finding the structure within chaos. By breaking the problem down into blocks—the 'A' block, the 'C' block, and so on—we turned a daunting task into a simple, elegant calculation.

You have just mastered the art of lexicographical ranking. Keep this logic in your toolkit; whether it is permutations or probability, the ability to organize and count systematically is the hallmark of a true problem solver.

The final rank of the word **SACHIN** is **601**.

Animated Solution for Mathematics - Permutations and Combinations: If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Select Answer:

Visualized Solution

The Word SACHIN

Alphabetical Order

Word Structure

Words Starting with A

Permutations for A

Skipping Other Letters

Counting Skipped Words

Total Words Before S

Reaching the Letter S

Arranging Remaining Letters

The First Word with S

Final Rank

The Sigma Insight: Linear Permutations

The Lexicographical Journey

Phase 1

Phase 2

Phase 3

The Takeaway

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