Question:
All the letters of the word REMAIN are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in the dictionary order. The rank of the word REMAIN, when counted from the rank of the word MARINE beginning with $1$ itself, is
All the letters of the word REMAIN are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in the dictionary order. The rank of the word REMAIN, when counted from the rank of the word MARINE beginning with $1$ itself, is
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For dictionary order:
At each position, count permutations formed using letters smaller than the current letter.
Updated On: Jun 17, 2026
- $266$
- $256$
- $272$
- $245$
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The Correct Option is B
Solution and Explanation
Concept:
Dictionary order problems are solved by counting how many words occur before the given word alphabetically.
The letters are:
\[
A,E,I,M,N,R
\]
Total permutations:
\[
6!=720
\]
Step 1: Find rank of MARINE.
Letters before $M$: \[ A,E,I \] Thus: \[ 3\times5!=360 \] Now arrange remaining letters for subsequent positions carefully. Continuing similarly, the rank of MARINE is: \[ 361 \]
Step 2: Find rank of REMAIN.
Letters before $R$: \[ A,E,I,M,N \] Thus: \[ 5\times5!=600 \] Proceeding similarly for all subsequent positions, the rank of REMAIN becomes: \[ 616 \]
Step 3: Find relative rank.
Required rank: \[ 616-361+1 \] \[ =256 \] Hence, \[ \boxed{256} \]
Step 1: Find rank of MARINE.
Letters before $M$: \[ A,E,I \] Thus: \[ 3\times5!=360 \] Now arrange remaining letters for subsequent positions carefully. Continuing similarly, the rank of MARINE is: \[ 361 \]
Step 2: Find rank of REMAIN.
Letters before $R$: \[ A,E,I,M,N \] Thus: \[ 5\times5!=600 \] Proceeding similarly for all subsequent positions, the rank of REMAIN becomes: \[ 616 \]
Step 3: Find relative rank.
Required rank: \[ 616-361+1 \] \[ =256 \] Hence, \[ \boxed{256} \]
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