Step 1: Understanding the Concept:
This is a problem of permutations because the order of the letters matters in forming different words.
Since repetition is not allowed, the number of choices decreases for each subsequent position.
Step 2: Key Formula or Approach:
The number of arrangements (permutations) of \( n \) distinct objects taken \( r \) at a time is:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Alternatively, use the multiplication principle of counting.
Step 3: Detailed Explanation:
The word "MATH" has 4 distinct letters (M, A, T, H).
We need to form 3-letter words.
Number of ways to fill the 1st position = 4 (any of M, A, T, H).
Number of ways to fill the 2nd position = 3 (one letter is already used).
Number of ways to fill the 3rd position = 2 (two letters are already used).
Using the multiplication principle:
\[ \text{Total number of words} = 4 \times 3 \times 2 = 24 \]
Using the formula \( P(4, 3) \):
\[ P(4, 3) = \frac{4!}{(4-3)!} = \frac{4 \times 3 \times 2 \times 1}{1} = 24 \]
Step 4: Final Answer:
The total number of 3-letter words that can be formed is 24.