How many 3 letter words can be formed from the letters of the word CALCUTTA?

Answer

To solve the problem of how many 3-letter words can be formed from the letters of the word "CALCUTTA", we will break it down into two cases based on the repetition of letters.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies**:
The letters in "CALCUTTA" are C, A, L, U, T, with the frequencies:
- C: 2
- A: 2
- L: 1
- U: 1
- T: 2

Thus, we have the unique letters: C, A, L, U, T.

2. **Case 1: All Letters are Unique**:
In this case, we will consider the unique letters only. The unique letters are C, A, L, U, T (5 unique letters).

- We need to choose 3 letters from these 5 unique letters.
- The number of ways to choose 3 letters from 5 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

- Each selection of 3 letters can be arranged in \( 3! \) ways:
\[ 3! = 6 \]

- Therefore, the total number of words formed in this case is:
\[ 10 \times 6 = 60 \]

3. **Case 2: Two Letters are the Same and One is Different**:
In this case, we need to select one letter that appears twice and one letter that appears once.

- The letters that can be chosen as the repeating letter are C, A, and T (since they appear twice).
- The letters that can be chosen as the different letter are L, U, C, A, and T (but we cannot choose the same letter as the repeating one).

- For each choice of the repeating letter, we have the following combinations:
- If we choose C (repeating), we can choose from A, L, U, T (4 options).
- If we choose A (repeating), we can choose from C, L, U, T (4 options).
- If we choose T (repeating), we can choose from C, A, L, U (4 options).

- Thus, for each of the 3 choices of repeating letters, we have 4 choices for the different letter:
\[ 3 \text{ (choices for repeating letter)} \times 4 \text{ (choices for different letter)} = 12 \]

- Each selection can be arranged in \( \frac{3!}{2!} \) ways (since two letters are the same):
\[ \frac{3!}{2!} = 3 \]

- Therefore, the total number of words formed in this case is:
\[ 12 \times 3 = 36 \]

4. **Total Count**:
Finally, we add the results from both cases:
\[ 60 + 36 = 96 \]

Thus, the total number of 3-letter words that can be formed from the letters of the word "CALCUTTA" is **96**.