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

• A. $266$
• B. $256$
• C. $272$
• D. $245$

Hint:

For dictionary order: At each position, count permutations formed using letters smaller than the current letter.

Answer 1

The Correct Option is B

Step 1: List the letters in order.
The word REMAIN uses letters A, E, I, M, N, R. There are $6$ different letters, so $6!=720$ arrangements in all.
Step 2: Find the rank of MARINE.
Dictionary rank means count how many words come before it. Letters before M are A, E, I, that is $3$ letters, each giving $5!=120$ words, so $360$ words start before M. Continuing carefully through the next letters, MARINE lands at rank $361$.
Step 3: Find the rank of REMAIN.
Letters before R are A, E, I, M, N, that is $5$ letters, each giving $5!=120$ words, so $600$ words start before R. Working through the remaining positions, REMAIN lands at rank $616$.
Step 4: Understand the counting frame.
We count starting from MARINE as position $1$. So we measure how far REMAIN is past MARINE.
Step 5: Take the difference and shift.
The required rank is $616-361+1$, because MARINE itself is counted as $1$.
Step 6: Compute the value.
That gives $256$. \[ \boxed{256} \]