How many different three-letter words can be formed using the letters of the word HEXAGON, if no letter is repeated in a word?

Introduction / Context:
This problem tests basic permutation ideas. We are given a word with several distinct letters and asked to count how many different three-letter arrangements or “words” can be formed if repetition of letters is not allowed. Such questions are very common in aptitude exams and directly use the fundamental principle of counting and the concept of permutations.

Given Data / Assumptions:

The word HEXAGON has 7 distinct letters: H, E, X, A, G, O, N.
We want to form three-letter words using these letters.
No letter may be repeated within a single three-letter word.
The order of letters matters. For example, HEG and GEH are considered different words.

Concept / Approach:
Whenever we arrange r distinct objects chosen from n distinct objects, and the order is important, we use permutations. The number of permutations is nPr, which equals n * (n - 1) * (n - 2) ... for r factors. Here n = 7 and r = 3. So the count is 7 * 6 * 5.

Step-by-Step Solution:
Step 1: Count available letters. HEXAGON contains 7 distinct letters, so n = 7.Step 2: We need three-letter words with no repetition, so r = 3.Step 3: For the first position, we can choose any of the 7 letters.Step 4: After choosing one letter, 6 letters remain for the second position.Step 5: After choosing two letters, 5 letters remain for the third position.Step 6: Total number of words = 7 * 6 * 5 = 210.

Verification / Alternative check:
We can also use the permutation notation directly: 7P3 = 7 * 6 * 5 = 210. This matches our stepwise reasoning, so the result is consistent.

Why Other Options Are Wrong:
120 assumes 5 * 4 * 3, as if only 5 letters were available or repetition rules were misunderstood.
160 is not a product of three consecutive integers from 7 and does not match any correct permutation count here.
200 is close to 210 but still incorrect and often appears as a trap when candidates miscalculate one of the factors.

Common Pitfalls:
Candidates often confuse permutations with combinations and may attempt to use only 7C3, which would ignore ordering. Others forget that the letters of HEXAGON are all distinct and may wrongly subtract for repeated letters that do not exist. Another mistake is to stop after computing 7 * 6 and forget to multiply by 5 for the third position.

Final Answer:
The number of different three-letter words that can be formed is 210.