JEE Advanced 2007
LEVELJEE Main
Visualized Solution
The Sigma Insight: Linear Permutations
Analyzing the Setup
To find the rank of the word in a dictionary, we first identify the letters present in the word: .
Sorting these letters alphabetically gives us the sequence: . This sorted list serves as our guide for the dictionary order.
The Systematic Search
Counting the Blocks
We are looking for the word . In a dictionary, words are ordered alphabetically by their first letter, then their second, and so on.
Since our target starts with , we must first count all words that start with prefixes appearing before . Based on our sorted list, these prefixes are , , , and .
For the block, we fix and in the first two positions. We are left with four letters: . Since these are all distinct, the number of arrangements is:
For the block, we fix and . The remaining letters are . These are four distinct letters, so the number of arrangements is:
For the block, we fix and . The remaining letters are . These are four distinct letters, so the number of arrangements is:
For the block, we fix and . The remaining letters are . These are four distinct letters, so the number of arrangements is:
The Final Reveal
Reaching COCHIN
We have now counted all the words that precede the block. The total number of words appearing before the block is:
Now, we arrive at the block. The remaining letters are . Arranging these alphabetically, we get .
This confirms that the very first word in the block is . Since our target word is , we have arrived at the word itself.
There are no other words starting with that come before it. Therefore, the total number of words appearing before is .
The rank of the word is .
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