(i) Find the number of different words formed by using all the letters of the word, 'INSTITUTION'.
In how many of them
(ii) are the three T's together ?
(iii) are the first two letters the two N's?

Explanation

To solve the problem, we will break it down into three parts as per the question.

### Part (i): Find the number of different words formed by using all the letters of the word 'INSTITUTION'.

1. **Count the letters in 'INSTITUTION'**:
- The word has 11 letters in total.
- The breakdown of the letters is as follows:
- I: 3 times
- N: 2 times
- S: 1 time
- T: 3 times
- U: 1 time
- O: 1 time

2. **Use the formula for permutations of multiset**:
The formula for the number of distinct permutations of a multiset is given by:
\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]
where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct items.

3. **Apply the formula**:
Here, \( n = 11 \) (total letters), \( n_1 = 3 \) (for I), \( n_2 = 2 \) (for N), \( n_3 = 3 \) (for T), \( n_4 = 1 \) (for S), \( n_5 = 1 \) (for U), \( n_6 = 1 \) (for O).
\[ \text{Number of distinct words} = \frac{11!}{3! \times 2! \times 3! \times 1! \times 1! \times 1!} \]

4. **Calculate the factorials**:
- \( 11! = 39916800 \)
- \( 3! = 6 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

5. **Substitute and simplify**:
\[ \text{Number of distinct words} = \frac{39916800}{6 \times 2 \times 6 \times 1 \times 1 \times 1} = \frac{39916800}{72} = 554400 \]

### Part (ii): Find the number of words where the three T's are together.

1. **Treat the three T's as a single unit**:
- When the three T's are together, we can treat them as one letter (let's call it TTT).
- This gives us the new set of letters: I, I, I, N, N, S, U, O, TTT (total of 9 letters).

2. **Count the frequency of the new letters**:
- I: 3 times
- N: 2 times
- S: 1 time
- U: 1 time
- O: 1 time
- TTT: 1 time

3. **Use the permutations formula again**:
\[ \text{Number of distinct words with T's together} = \frac{9!}{3! \times 2! \times 1! \times 1! \times 1!} \]

4. **Calculate the factorials**:
- \( 9! = 362880 \)

5. **Substitute and simplify**:
\[ \text{Number of distinct words with T's together} = \frac{362880}{6 \times 2 \times 1 \times 1 \times 1} = \frac{362880}{12} = 30240 \]

### Part (iii): Find the number of words where the first two letters are the two N's.

1. **Fix the first two letters as N, N**:
- This leaves us with the letters: I, I, I, T, T, T, S, U, O (total of 9 letters).

2. **Count the frequency of the remaining letters**:
- I: 3 times
- T: 3 times
- S: 1 time
- U: 1 time
- O: 1 time

3. **Use the permutations formula**:
\[ \text{Number of distinct words with first two letters N, N} = \frac{9!}{3! \times 3! \times 1! \times 1! \times 1!} \]

4. **Calculate the factorials**:
- \( 9! = 362880 \)

5. **Substitute and simplify**:
\[ \text{Number of distinct words with first two letters N, N} = \frac{362880}{6 \times 6 \times 1 \times 1 \times 1} = \frac{362880}{36} = 10080 \]

### Final Answers:
(i) 554400
(ii) 30240
(iii) 10080