How many 4-letter words can be formed using letters of the word 'VIDYAMANDIR' such that the word always includes a letter 'I '.

Options

A. 1320
B. 1546
C. 1248
D. 1456

Answer

To solve the problem of how many 4-letter words can be formed using the letters of the word "VIDYAMANDIR" that always include the letter 'I', we can break it down into cases based on the number of 'I's included in the 4-letter words.

### Step-by-Step Solution:

1. **Identify the Letters**:
The letters in "VIDYAMANDIR" are V, I, D, Y, A, M, A, N, D, I, R. The unique letters are V, I, D, Y, A, M, N, R, with 'I' appearing twice and 'A' appearing twice.

2. **Case 1: 2 I's in the word**:
- **Sub-case 1.1: Both I's are the same**:
- Choose 2 more letters from {V, D, Y, A, M, N, R} (7 options).
- The number of ways to choose 2 letters from 7 is \( \binom{7}{2} = 21 \).
- The arrangement of the letters (I, I, X, Y) is given by \( \frac{4!}{2!} = 12 \).
- Total for this sub-case: \( 21 \times 12 = 252 \).

- **Sub-case 1.2: Both I's are different**:
- This case is not possible since we only have two 'I's.

3. **Case 2: 1 I in the word**:
- **Sub-case 2.1: 2 same letters + 1 different letter**:
- Choose 2 same letters from {A, D} (2 options).
- Choose 1 letter from the remaining letters (6 options).
- The arrangement of the letters (I, X, X, Y) is given by \( \frac{4!}{2!} = 12 \).
- Total for this sub-case: \( 2 \times 6 \times 12 = 144 \).

- **Sub-case 2.2: All different letters**:
- Choose 3 letters from {V, D, Y, A, M, N, R} (7 options).
- The number of ways to choose 3 letters from 7 is \( \binom{7}{3} = 35 \).
- The arrangement of the letters (I, X, Y, Z) is given by \( 4! = 24 \).
- Total for this sub-case: \( 35 \times 24 = 840 \).

4. **Combine All Cases**:
- Total from Case 1: 252 (only sub-case 1.1).
- Total from Case 2: 144 (sub-case 2.1) + 840 (sub-case 2.2) = 984.
- Overall total: \( 252 + 984 = 1236 \).

### Final Answer:
The total number of 4-letter words that can be formed using the letters of "VIDYAMANDIR" that always include the letter 'I' is **1236**.