Five distinct letters k, l, m, n and o are available. Using any three of these letters at a time, and not repeating any letter within a word, how many different three letter words (with or without meaning) can be formed?

Introduction / Context:
This question checks understanding of permutations when we form words or arrangements from a subset of available distinct letters. The focus is on recognising that order matters in forming three letter words and that each letter can be used at most once in any particular word. This is a standard aptitude topic under permutations and combinations and often appears in competitive exams.

Given Data / Assumptions:

• Available letters are k, l, m, n and o, so there are 5 distinct letters.
• We need to form words of exactly three letters.
• No letter is repeated within a word.
• The words need not have any dictionary meaning; they are simply arrangements.

Concept / Approach:
Since the words are three letter arrangements where order matters and repetition is not allowed, we use permutations. Specifically, we are finding the number of permutations of 5 distinct letters taken 3 at a time. This is denoted by 5P3, and the formula for nPr is n! / (n - r)!. We will calculate the total by multiplying the number of choices for each position in the word step by step.

Step-by-Step Solution:
Step 1: For the first letter of the word, we can choose any of the 5 letters, so there are 5 choices.Step 2: For the second letter, one letter has already been used, so 4 letters remain, giving 4 choices.Step 3: For the third letter, two letters are already used, so 3 letters remain, giving 3 choices.Step 4: Multiply the choices: total number of words = 5 * 4 * 3 = 60.Step 5: Therefore, the required number of three letter words is 60.
Verification / Alternative check:
The result can also be confirmed using the permutation formula 5P3 = 5! / (5 - 3)! = 5! / 2! = (5 * 4 * 3 * 2 * 1) / (2 * 1) = 5 * 4 * 3 = 60. Both the direct counting method and the formula give the same value, so the answer is consistent.

Why Other Options Are Wrong:
120 would be 5! and would correspond to arranging all five letters, not just three at a time. 240 is an overestimate and does not correspond to any standard arrangement count in this scenario. 30 would come from 5C3 and would count combinations where order does not matter, which is not appropriate because different orders like klm and mlk are distinct words.

Common Pitfalls:
A common mistake is to use combinations instead of permutations, thereby forgetting that different orders of letters give different words. Another mistake is to accidentally allow repetition, which would change the calculation to 5^3. Careful reading of the condition "without repetition of alphabets" avoids such errors.

Final Answer:
The number of three letter words that can be formed is 60.