The letters of the word 'INDIA' are arranged as in a dictionary. What are the 1st, 13th, 25th, 49th and 60th words?

Explanation

To find the 1st, 13th, 25th, 49th, and 60th words formed by the letters of the word 'INDIA' when arranged in alphabetical order, we will follow these steps:

### Step 1: Identify the letters and their frequency
The letters in 'INDIA' are: I, N, D, I, A.
- Frequencies:
- A: 1
- D: 1
- I: 2
- N: 1

### Step 2: Arrange the letters in alphabetical order
The alphabetical order of the letters is: A, D, I, I, N.

### Step 3: Calculate the total number of unique arrangements
The total number of arrangements of the letters can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{5!}{2!} = \frac{120}{2} = 60 \]
So, there are 60 unique arrangements of the letters in 'INDIA'.

### Step 4: Find the 1st word
To find the first word, we start with the first letter in alphabetical order:
- The first letter is A.
- The remaining letters are D, I, I, N.
- Arranging these gives us: **A D I I N**.

Thus, the 1st word is **AD I I N**.

### Step 5: Find the 13th word
Next, we need to find the 13th word.
- The first 12 words start with A.
- The next letter after A is D, and we need to arrange the remaining letters I, I, N.
- The arrangements starting with D are: D, A, I, I, N.
- The number of arrangements starting with D is also \( \frac{4!}{2!} = 12 \).

Thus, the 13th word starts with D:
- The next arrangement after A is D.
- The remaining letters are A, I, I, N.
- Arranging these gives us: **D A I I N**.

Thus, the 13th word is **D A I I N**.

### Step 6: Find the 25th word
For the 25th word:
- The first 24 words consist of those starting with A (12 words) and those starting with D (12 words).
- The next letter after D is I.
- The arrangements starting with I are: I, A, D, I, N.
- The number of arrangements starting with I is \( \frac{4!}{1!} = 24 \).

Thus, the 25th word starts with I:
- The next arrangement after D is I.
- The remaining letters are A, D, I, N.
- Arranging these gives us: **I A D I N**.

Thus, the 25th word is **I A D I N**.

### Step 7: Find the 49th word
For the 49th word:
- The first 48 words consist of those starting with A (12), D (12), and I (24).
- The next letter after I is N.
- The arrangements starting with N are: N, A, D, I, I.
- The number of arrangements starting with N is \( \frac{4!}{2!} = 12 \).

Thus, the 49th word starts with N:
- The next arrangement after I is N.
- The remaining letters are A, D, I, I.
- Arranging these gives us: **N A D I I**.

Thus, the 49th word is **N A D I I**.

### Step 8: Find the 60th word
For the 60th word:
- The last arrangement will be the one that starts with the largest letter.
- The largest letter is N.
- The remaining letters are I, I, D, A.
- Arranging these gives us: **N I I D A**.

Thus, the 60th word is **N I I D A**.

### Summary of Results
- 1st word: **A D I I N**
- 13th word: **D A I I N**
- 25th word: **I A D I N**
- 49th word: **N A D I I**
- 60th word: **N I I D A**