In a survey it was found that `21` people liked product `A`, `26` liked product `B` and `29` liked product `C`. If `14` people liked products `A` and `B`, `12` people liked products `C` and `A`, `14` people liked products `B` and `C` and `8` liked all the three products. Find how many liked products `C` only?

Explanation

`n(A) = 21 and n(B)=26`
Number of people who like product A and B both
`n(A cupB)=n(A)+n(B)-n(A capB)`
`=21+26-14=33`
Now `n(C)=29`
Number of people who like product B and C both
`n(B cap C)=14`
Number of people who like product C and A both
`n(A cap C)=12`
Number of people who like product A,B and C, all
`n(A capB cap C)=8`
Then, number of people who like any one of the one product
`n(AcupBcupC)=n(A)+n(B)+n(C)-n(AcapB)`
`-n(BcapC)-n(AcapC)+n(A cap Bcap C)`
`= 21+26+29-14-14-12 +8`
`=44`
`becausen(A cupBcupC)=44and n(A cupB)=33`

Number of people who like only product C
`= n(A cupBcupC)-n(A cupB)=44-33=11`
Therefore, number of people who like only product
`C = 11`.