Animated Solution for Mathematics - Permutations and Combinations: If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

Unlocking the Rank of THAMS

Welcome, future engineer! Today, we are not just solving a permutation problem; we are embarking on a journey through the lexicographical order of the English language.

Imagine you are standing in a vast library, holding the word $\text{MATHS}$. Your task is to find the exact serial number of the word $\text{THAMS}$ if all possible permutations of $\text{MATHS}$ were arranged in a dictionary. This is a classic JEE Advanced challenge, and it requires a systematic, almost algorithmic mindset.

The Foundation of Order

Before we touch any math, we must establish the ground rules. A dictionary is strictly alphabetical.

Our first step is to sort the letters of $\text{MATHS}$ in ascending order: $\text{A, H, M, S, T}$. This is our master key. Every word we form will follow this hierarchy.

Our target word is $\text{THAMS}$. We need to see how many words precede it.

The Great Bypass

We want to reach words starting with $\text{T}$. But in the dictionary, words starting with $\text{A, H, M,}$ and $\text{S}$ will appear long before we ever see a word starting with $\text{T}$. We need to count these blocks.

If we fix $\text{A}$ in the first position, we have $4$ remaining slots to fill with the letters $\text{H, M, S, T}$. The number of ways to arrange these is:

This means there are $24$ words starting with $\text{A}$. The same logic applies to $\text{H, M,}$ and $\text{S}$.

Since each of these letters precedes $\text{T}$, we must count all of them:

We have successfully bypassed $96$ words that come before our target block.

The Descent into the T-Block

Now, we arrive at the words starting with $\text{T}$. Our target word $\text{THAMS}$ starts with $\text{T}$, so we are finally in the right neighborhood!

We lock $\text{T}$ in the first position. Now, we look at the second position. The available letters are $\text{A, H, M, S}$.

Alphabetically, $\text{A}$ comes first. So, we consider words starting with $\text{TA}$. Since our target word is $\text{THAMS}$, any word starting with $\text{TA}$ comes before $\text{THAMS}$.

We fix $\text{T}$ and $\text{A}$. We have $3$ slots remaining, which gives us:

Our running total is now $96+6=102$.

The Final Match

We move to the next available letter for the second position, which is $\text{H}$. Our word starts with $\text{TH}$. This is a match!

We lock $\text{H}$ in the second position. Now, we look at the third position. The remaining letters are $\text{A, M, S}$. Alphabetically, $\text{A}$ comes first. We place $\text{A}$.

The word starts with $\text{THA}$. This is also a match! We lock $\text{A}$.

For the fourth position, we have $\text{M}$ and $\text{S}$ left. $\text{M}$ comes before $\text{S}$. We place $\text{M}$. The word starts with $\text{THAM}$. This is a match!

Finally, we are left with only $\text{S}$ for the last position. The word is $\text{THAMS}$. We have arrived at our destination.

We have counted $96$ words starting with $\text{A, H, M, S}$, plus $6$ words starting with $\text{TA}$, and finally, we add $1$ for the word $\text{THAMS}$ itself.

The total rank is:

It is a beautiful, logical progression. Remember, in these problems, patience is your greatest tool. Don't rush the counting, and always respect the alphabetical order. You have mastered the dictionary logic!