If the letters of the word VERMA are written in all possible arrangements in dictionary order, what is the rank (position) of the word VERMA in this ordered list?

Introduction / Context:
This question examines your understanding of lexicographic (dictionary) order and permutations of distinct letters. The rank of a particular arrangement tells you how many valid permutations come before it when all are listed from smallest to largest in alphabetical order, which is a classic type of reasoning problem in aptitude tests.

Given Data / Assumptions:

• The word is VERMA.
• The letters are all distinct: V, E, R, M, A.
• Dictionary order means standard alphabetical order with letters arranged from A to Z.
• We must count all permutations of these letters that appear before VERMA and then add one to obtain its rank.

Concept / Approach:
First list the letters in alphabetical order: A, E, M, R, V. Then, position by position, count how many permutations begin with a letter smaller than the corresponding letter of VERMA. For each such smaller starting choice, the remaining letters can permute freely. The total count of these permutations gives the number of words before VERMA.

Step-by-Step Solution:
Alphabetic order of letters: A, E, M, R, V.First letter V: smaller letters are A, E, M, R (4 letters).For each such choice, the remaining 4 letters can be arranged in 4! = 24 ways. Contribution: 4 * 24 = 96.Now fix V. Remaining letters: A, E, M, R in alphabetical order.Second letter of VERMA is E. Smaller among remaining are A only (1 letter). The remaining 3 letters can arrange in 3! = 6 ways. Add 6 to total: 96 + 6 = 102.Fix V, E. Remaining letters: A, M, R. Third letter is R. Smaller letters are A and M (2 letters). Each yields 2! = 2 arrangements for the last two letters. Contribution: 2 * 2 = 4. Total so far: 106.Fix V, E, R. Remaining letters: A, M. Fourth letter is M. Smaller letter is A (1 letter). With A in fourth position, last letter must be M, so 1 arrangement. New total: 107.The word VERMA itself comes next, so its rank is 107 + 1 = 108.

Verification / Alternative check:
You can cross check by quickly listing a few final permutations around VERMA to see that exactly 107 distinct permutations precede it.The stepwise positional counting is the standard and reliable method.

Why Other Options Are Wrong:
117 and 180 are larger ranks that would require more permutations before VERMA than actually exist.810 is far too large, exceeding the total permutations 5! = 120.

Common Pitfalls:
Forgetting to keep letters in strict alphabetical order at each step when counting smaller possibilities.Not multiplying by factorial of remaining letters after fixing earlier choices.Assuming that the rank is simply the numeric value or position of some letter without doing full combinatorial counting.

Final Answer:
The rank of the word VERMA is 108.