If all the letters of the word 'AGAIN' be arranged as in a dictionary, then fiftieth word is

Options

A. NAAGI
B. NAGAI
C. NAAIG
D. NAIAG

Answer

To find the 50th word formed by arranging the letters of the word "AGAIN" in dictionary order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters in "AGAIN" are A, A, G, I, N. When arranged in alphabetical order, they are:
- A, A, G, I, N

### Step 2: Calculate the number of arrangements starting with each letter
1. **Starting with A**:
- Remaining letters: A, G, I, N
- The number of arrangements = 4! / 2! (since there are 2 A's)
- = 24 / 2 = 12 arrangements

2. **Starting with G**:
- Remaining letters: A, A, I, N
- The number of arrangements = 4! / 2! (since there are 2 A's)
- = 24 / 2 = 12 arrangements

3. **Starting with I**:
- Remaining letters: A, A, G, N
- The number of arrangements = 4! / 2! (since there are 2 A's)
- = 24 / 2 = 12 arrangements

### Step 3: Total arrangements counted so far
- Arrangements starting with A: 12
- Arrangements starting with G: 12
- Arrangements starting with I: 12

Total = 12 + 12 + 12 = 36 arrangements

### Step 4: Start with N
Now, we move to arrangements starting with N since we have only counted 36 arrangements so far.

1. **Starting with N**:
- Remaining letters: A, A, G, I
- The number of arrangements = 4! / 2! (since there are 2 A's)
- = 24 / 2 = 12 arrangements

### Step 5: Count the arrangements starting with N
Now we can count the arrangements starting with N:
- The first arrangement starting with N will be: NAAIG
- The second arrangement will be: NAAIG
- The third arrangement will be: NAAGI
- The fourth arrangement will be: NAIA
- The fifth arrangement will be: NAGAI
- The sixth arrangement will be: NAGIA
- The seventh arrangement will be: NAIAG
- The eighth arrangement will be: NAIAG
- The ninth arrangement will be: NAGAI
- The tenth arrangement will be: NAGIA
- The eleventh arrangement will be: NAIAG
- The twelfth arrangement will be: NAIAG

### Step 6: Identify the 50th word
Since we have already counted 48 arrangements (12 starting with A, 12 starting with G, and 12 starting with I), the 49th arrangement will start with N. The 50th arrangement will be the first arrangement starting with N, which is:
- NAAIG

### Final Answer
Thus, the 50th word in the dictionary arrangement of the letters of "AGAIN" is **NAAIG**.

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