How many different words can be formed by taking four letters out of the letters of word 'AGAIN' if each word has to start with A?

Options

A. 6
B. 12
C. 24
D. None

Answer

To solve the problem of how many different words can be formed by taking four letters out of the letters of the word "AGAIN," with the condition that each word must start with 'A', we can follow these steps:

### Step 1: Identify the letters in "AGAIN"
The word "AGAIN" consists of the letters: A, G, A, I, N.

### Step 2: Fix the first letter
Since each word must start with 'A', we fix 'A' as the first letter. This leaves us with the letters: A, G, I, N (the second 'A' can still be used).

### Step 3: Choose the remaining letters
We need to choose 3 more letters from the remaining letters: A, G, I, N.

### Step 4: List the possible combinations
The possible combinations of letters we can choose (considering we have one 'A' fixed) are:
1. A, G, I
2. A, G, N
3. A, I, N
4. G, I, N

### Step 5: Calculate the arrangements
Now we will arrange these combinations.
- For the combination A, G, I: The letters are A, A, G, I. The number of arrangements is calculated using the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots} \]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

Here, we have:
- Total letters = 4 (A, A, G, I)
- The frequency of A = 2, G = 1, I = 1.

Therefore, the number of arrangements for A, A, G, I is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

- For the combination A, G, N: The letters are A, A, G, N. Similarly, the number of arrangements is:
\[ \frac{4!}{2!} = 12 \]

- For the combination A, I, N: The letters are A, A, I, N. The number of arrangements is:
\[ \frac{4!}{2!} = 12 \]

- For the combination G, I, N: The letters are A, G, I, N. The number of arrangements is:
\[ 4! = 24 \]

### Step 6: Total arrangements
Now we sum all the arrangements:
\[ 12 + 12 + 12 + 24 = 60 \]

### Final Answer
Thus, the total number of different words that can be formed by taking four letters out of the letters of the word "AGAIN" that start with 'A' is **60**.

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