(i) Find the number of words formed with the letters of the word 'MATHEMATICS' in which vowels never occur together.
(ii) Find the number of words formed with the letters of the word'CHANDIGARH'. In now many words, both 'A' will not be together?

Answer

To solve the given problems step by step, we will break down each part of the question and provide a clear solution.

### Part (i): Find the number of words formed with the letters of the word 'MATHEMATICS' in which vowels never occur together.

**Step 1: Identify the letters and their frequencies.**
The word 'MATHEMATICS' consists of the following letters:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1

Total letters = 11 (2M, 2A, 2T, H, E, I, C, S)

**Step 2: Calculate the total arrangements of the letters.**
The total arrangements of the letters can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{11!}{2! \times 2! \times 2!} \]

**Step 3: Calculate the arrangements when vowels are together.**
The vowels in 'MATHEMATICS' are A, A, E, I. If we treat all vowels as a single entity (block), we have:
- Vowel block: (AAEI)
- Consonants: M, M, T, T, H, C, S

Now, we have the following letters to arrange:
- Vowel block (1)
- M: 2
- T: 2
- H: 1
- C: 1
- S: 1

This gives us a total of 8 entities (1 vowel block + 7 consonants).

The arrangements are:
\[ \text{Arrangements with vowels together} = \frac{8!}{2! \times 2!} \]
The vowel block (AAEI) can also be arranged among themselves:
\[ \text{Arrangements of vowels} = \frac{4!}{2!} \]

**Step 4: Calculate the total arrangements with vowels together.**
Thus, the total arrangements with vowels together is:
\[ \text{Total with vowels together} = \frac{8!}{2! \times 2!} \times \frac{4!}{2!} \]

**Step 5: Calculate the arrangements where vowels do not occur together.**
To find the arrangements where vowels do not occur together, we subtract the arrangements with vowels together from the total arrangements:
\[ \text{Vowels not together} = \text{Total arrangements} - \text{Total with vowels together} \]

### Part (ii): Find the number of words formed with the letters of the word 'CHANDIGARH' in which both 'A's will not be together.

**Step 1: Identify the letters and their frequencies.**
The word 'CHANDIGARH' consists of the following letters:
- C: 1
- H: 2
- A: 2
- N: 1
- D: 1
- I: 1
- G: 1
- R: 1
- Total letters = 10

**Step 2: Calculate the total arrangements of the letters.**
The total arrangements of the letters can be calculated as:
\[ \text{Total arrangements} = \frac{10!}{2! \times 2!} \]

**Step 3: Calculate the arrangements when both A's are together.**
If we treat both A's as a single entity (block), we have:
- A block: (AA)
- Other letters: C, H, H, N, D, I, G, R

Now we have 9 entities (1 A block + 8 other letters).

The arrangements are:
\[ \text{Arrangements with A's together} = \frac{9!}{2!} \]

**Step 4: Calculate the arrangements where A's do not occur together.**
To find the arrangements where both A's do not occur together, we subtract the arrangements with A's together from the total arrangements:
\[ \text{A's not together} = \text{Total arrangements} - \text{Arrangements with A's together} \]

### Final Answers
1. For the word 'MATHEMATICS', the number of arrangements where vowels do not occur together is calculated using the above steps.
2. For the word 'CHANDIGARH', the number of arrangements where both A's do not occur together is calculated similarly.