A word has 4 identical letters and rest all are distinct letters. If the total number of words that can be made with the letters of the word be 210, then the total number of different letters in the word is equal to

Options

A. 3
B. 5
C. 4
D. 7

Answer

To solve the problem, we need to determine the total number of different letters in a word that consists of 4 identical letters and the rest distinct letters, given that the total number of arrangements of these letters is 210.

### Step-by-step Solution:

1. **Understanding the Problem**:
- We have a word with 4 identical letters (let's denote them as 'A') and the rest are distinct letters.
- Let the total number of letters in the word be \( n \).
- The total number of arrangements of these letters is given as 210.

2. **Using the Formula for Arrangements**:
- The formula for the number of arrangements of letters where some letters are identical is given by:
\[ \text{Number of arrangements} = \frac{n!}{k!} \]
where \( n \) is the total number of letters and \( k \) is the number of identical letters.
- In our case, \( k = 4 \) (the identical letters).

3. **Setting Up the Equation**:
- Thus, we can write:
\[ \frac{n!}{4!} = 210 \]
- Rearranging gives:
\[ n! = 210 \times 4! \]

4. **Calculating \( 4! \)**:
- We know that \( 4! = 24 \).
- Therefore, substituting this value in gives:
\[ n! = 210 \times 24 \]

5. **Calculating \( 210 \times 24 \)**:
- Now, calculate \( 210 \times 24 \):
\[ 210 \times 24 = 5040 \]
- So, we have:
\[ n! = 5040 \]

6. **Finding \( n \)**:
- We need to find \( n \) such that \( n! = 5040 \).
- Checking factorial values:
- \( 1! = 1 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)
- \( 6! = 720 \)
- \( 7! = 5040 \) (This is our value)

Thus, \( n = 7 \).

7. **Calculating the Number of Different Letters**:
- We have 4 identical letters and the total number of letters is 7.
- The number of distinct letters is:
\[ \text{Number of different letters} = n - 4 = 7 - 4 = 3 \]

### Final Answer:
The total number of different letters in the word is **3**.