All the letters of the word 'AGAIN' be arranged and the words thus formed are known as 'Simple Words'. Further two new types of words are defined as follows:
(i) Smart word: all the letters of the word 'AGAIN' are being used, but vowels can be repeated as many times as we need.
(ii) Dull word: All the letters of the word 'AGAIN' are being used, but consonants can be repeated as many times as we need.
Q. Number of 7 letter smart words is a. 1500 b. 1050 c. 1005 d. 150

Options

A. 1500
B. 1050
C. 1005
D. 150

Explanation

To solve the problem of finding the number of 7-letter "smart words" that can be formed using the letters of the word "AGAIN", we will follow these steps:

### Step 1: Identify the letters and their types
The letters in the word "AGAIN" are:
- Vowels: A, A, I (with A repeating)
- Consonants: G, N

### Step 2: Determine the total letters needed
We need to form a 7-letter word using all the letters from "AGAIN". Since we only have 5 letters (A, G, A, I, N), we need to repeat some letters to make a total of 7 letters.

### Step 3: Identify cases for vowel repetition
Since we can repeat vowels, we can consider three cases for filling the additional two letters:
1. Case 1: Both additional letters are A (A, A)
2. Case 2: One additional letter is A and the other is I (A, I)
3. Case 3: Both additional letters are I (I, I)

### Step 4: Calculate permutations for each case

#### Case 1: Two A's added (A, A)
- Letters: A, A, A, G, I, N (total 7 letters)
- Arrangement:
\[ \text{Permutations} = \frac{7!}{4!} = \frac{5040}{24} = 210 \]

#### Case 2: One A and one I added (A, I)
- Letters: A, A, I, G, I, N (total 7 letters)
- Arrangement:
\[ \text{Permutations} = \frac{7!}{2! \cdot 3!} = \frac{5040}{2 \cdot 6} = \frac{5040}{12} = 420 \]

#### Case 3: Two I's added (I, I)
- Letters: A, A, A, G, N, I (total 7 letters)
- Arrangement:
\[ \text{Permutations} = \frac{7!}{3! \cdot 2!} = \frac{5040}{6 \cdot 2} = \frac{5040}{12} = 420 \]

### Step 5: Sum the total permutations from all cases
Now, we add the permutations from all three cases:
\[ \text{Total} = 210 + 420 + 420 = 1050 \]

### Final Answer
The total number of 7-letter smart words that can be formed is **1050**.