Step 1: Understanding the Concept:
This is a problem involving Permutations, which is a branch of combinatorics.
Permutation refers to the number of different ways a set of items can be ordered or arranged.
When we arrange letters in a word, the position of each letter matters.
If all the items (letters) in the set are distinct, the calculation is straightforward using the factorial function.
Key Formula or Approach:
The number of permutations of \( n \) distinct objects is given by \( n! \) (n-factorial).
\[ n! = n \times (n-1) \times (n-2) \times \dots \times 1 \]
Step 2: Detailed Explanation:
The given word is "CAT".
Let's count the number of letters in the word:
1. C
2. A
3. T
The total number of letters (\( n \)) is 3.
Notice that all letters are unique (there are no repeated letters).
Therefore, the number of ways to arrange these 3 letters is \( 3! \).
Calculate the factorial:
\[ 3! = 3 \times 2 \times 1 \]
\[ 3 \times 2 = 6 \]
\[ 6 \times 1 = 6 \]
To visualize this, we can list all possible arrangements:
1. CAT
2. CTA
3. ACT
4. ATC
5. TCA
6. TAC
There are a total of 6 distinct arrangements.
If the word was "AAT", we would have to divide by \( 2! \) because of the repeated 'A's, but since "CAT" has all distinct letters, the answer is exactly 6.
Step 3: Final Answer:
The letters of the word "CAT" can be arranged in 6 different ways.
This corresponds to Option (B).