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 step by step, we will use the principle of inclusion-exclusion and Venn diagrams.

### Given Data:
- Let \( n(A) = 21 \) (persons who liked product A)
- Let \( n(B) = 26 \) (persons who liked product B)
- Let \( n(C) = 29 \) (persons who liked product C)
- Let \( n(A \cap B) = 14 \) (persons who liked both A and B)
- Let \( n(A \cap C) = 12 \) (persons who liked both A and C)
- Let \( n(B \cap C) = 13 \) (persons who liked both B and C)
- Let \( n(A \cap B \cap C) = 8 \) (persons who liked all three products)

### (i) Find the number of persons who liked product C only.

To find the number of persons who liked only product C, we can use the formula:
\[ n(C \text{ only}) = n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

Substituting the values:
\[ n(C \text{ only}) = 29 - 12 - 13 + 8 \]

Calculating this step by step:
1. \( 29 - 12 = 17 \)
2. \( 17 - 13 = 4 \)
3. \( 4 + 8 = 12 \)

Thus, the number of persons who liked product C only is \( 12 \).

### (ii) Find the number of persons who liked products A and B but not C.

To find the number of persons who liked both A and B but not C, we can use the formula:
\[ n(A \cap B \text{ but not } C) = n(A \cap B) - n(A \cap B \cap C) \]

Substituting the values:
\[ n(A \cap B \text{ but not } C) = 14 - 8 \]

Calculating this:
1. \( 14 - 8 = 6 \)

Thus, the number of persons who liked products A and B but not C is \( 6 \).

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