In a survery, it was found that `21` persons liked product `A`, `26` liked product `B` and `29` liked product `C`. If `14` persons liked products `A` and `B`, `12` liked products `C` and `A`, `13` persons liked products `B` and `C` and `8` liked all the three products then
`(i)` Find the number of persons who liked the product `C` only
`(ii)` The number of persons who like the products `A` and `B` but not `C`

Answer

To solve the problem, we will use a Venn diagram to represent the relationships between the persons who liked products A, B, and C. Let's denote the number of persons who liked:

- Product A = 21
- Product B = 26
- Product C = 29
- Both A and B = 14
- Both A and C = 12
- Both B and C = 13
- All three products (A, B, and C) = 8

### Step 1: Fill in the Venn Diagram

1. **All three products (A, B, C)**: The number of persons who liked all three products is given as 8. We place this in the center where all three circles overlap.

2. **A and B only**: The number of persons who liked both A and B is 14. Since 8 liked all three products, the number who liked only A and B is:
\[ (A \cap B) - (A \cap B \cap C) = 14 - 8 = 6 \]
We place 6 in the section where only A and B overlap.

3. **A and C only**: The number of persons who liked both A and C is 12. Again, since 8 liked all three products, the number who liked only A and C is:
\[ (A \cap C) - (A \cap B \cap C) = 12 - 8 = 4 \]
We place 4 in the section where only A and C overlap.

4. **B and C only**: The number of persons who liked both B and C is 13. Therefore, the number who liked only B and C is:
\[ (B \cap C) - (A \cap B \cap C) = 13 - 8 = 5 \]
We place 5 in the section where only B and C overlap.

5. **A only**: The total number of persons who liked product A is 21. We have already accounted for 6 (A and B only), 4 (A and C only), and 8 (A, B, and C). Thus, the number who liked only A is:
\[ A - (A \cap B) - (A \cap C) + (A \cap B \cap C) = 21 - 6 - 4 - 8 = 3 \]
We place 3 in the section for A only.

6. **B only**: The total number of persons who liked product B is 26. We have accounted for 6 (A and B only), 5 (B and C only), and 8 (A, B, and C). Thus, the number who liked only B is:
\[ B - (A \cap B) - (B \cap C) + (A \cap B \cap C) = 26 - 6 - 5 - 8 = 7 \]
We place 7 in the section for B only.

7. **C only**: The total number of persons who liked product C is 29. We have accounted for 4 (A and C only), 5 (B and C only), and 8 (A, B, and C). Thus, the number who liked only C is:
\[ C - (A \cap C) - (B \cap C) + (A \cap B \cap C) = 29 - 4 - 5 - 8 = 12 \]
We place 12 in the section for C only.

### Summary of the Venn Diagram
- Only A: 3
- Only B: 7
- Only C: 12
- A and B only: 6
- A and C only: 4
- B and C only: 5
- A, B, and C: 8

### Step 2: Answer the Questions

(i) **Find the number of persons who liked product C only**:
From our calculations, we found that the number of persons who liked only product C is **12**.

(ii) **The number of persons who like the products A and B but not C**:
From our calculations, we found that the number of persons who liked both A and B but not C is **6**.

### Final Answers
(i) The number of persons who liked product C only is **12**.
(ii) The number of persons who like products A and B but not C is **6**.