The number of different words that can be formed out of the letters of the word MORADABAD taken four at a time is

Options

A. 500
B. 600
C. 620
D. 626

Answer

To find the number of different words that can be formed from the letters of the word "MORADABAD" taken four at a time, we will analyze the letters and their frequencies, and then consider different cases based on the repetition of letters.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies:**
The word "MORADABAD" consists of the following letters:
- M: 1
- O: 1
- R: 1
- A: 3
- D: 2
- B: 1

Total letters = 9 (M, O, R, A, A, A, D, D, B)

2. **Case Analysis:**
We will consider different cases based on the number of repeated letters.

**Case 1: All four letters are different.**
- The different letters available are M, O, R, A, D, B (total 6 different letters).
- We can choose 4 letters from these 6 letters:
\[ \text{Number of ways to choose 4 letters} = \binom{6}{4} = 15 \]
- Each selection of 4 letters can be arranged in \(4!\) ways:
\[ 4! = 24 \]
- Therefore, the total for this case is:
\[ 15 \times 24 = 360 \]

**Case 2: 2 same and 2 different.**
- The letters that can be the same are A (2) or D (2).
- **Sub-case 2.1: A is the repeated letter.**
- Choose 2 different letters from M, O, R, D, B (5 choices):
\[ \binom{5}{2} = 10 \]
- Arrangements:
\[ \frac{4!}{2!} = 12 \]
- Total for this sub-case:
\[ 10 \times 12 = 120 \]

- **Sub-case 2.2: D is the repeated letter.**
- Choose 2 different letters from M, O, R, A, B (5 choices):
\[ \binom{5}{2} = 10 \]
- Arrangements:
\[ \frac{4!}{2!} = 12 \]
- Total for this sub-case:
\[ 10 \times 12 = 120 \]

- Therefore, total for Case 2:
\[ 120 + 120 = 240 \]

**Case 3: 3 same and 1 different.**
- The only letter that can be repeated 3 times is A.
- Choose 1 letter from M, O, R, D, B (5 choices):
\[ \binom{5}{1} = 5 \]
- Arrangements:
\[ \frac{4!}{3!} = 4 \]
- Total for this case:
\[ 5 \times 4 = 20 \]

**Case 4: 2 same and 2 same.**
- The only combination possible is A, A, D, D.
- Arrangements:
\[ \frac{4!}{2! \times 2!} = 6 \]
- Total for this case:
\[ 6 \]

3. **Total Count of Different Words:**
Now, we add the totals from all cases:
\[ \text{Total} = 360 + 240 + 20 + 6 = 626 \]

### Final Answer:
The number of different words that can be formed out of the letters of the word "MORADABAD" taken four at a time is **626**.