A news agent sells three papers: The Times, the Punch and the Herald. $70$ customers buy the Times, $60$ the Punch and $50$ the Herald. $17$ buy the Times and the Punch, $15$ the Punch and the Herald and $16$ the Herald and the Times, while $3$ customers buy all three papers. How many customers has he?

Full solution

Q. A news agent sells three papers: The Times, the Punch and the Herald.

70

customers buy the Times,

60

the Punch and

50

the Herald.

17

buy the Times and the Punch,

15

the Punch and the Herald and

16

the Herald and the Times, while

3

customers buy all three papers. How many customers has he?

Identify Customers per Paper: Identify the total number of customers buying each individual paper. $\newline$ The Times: $70$ customers $\newline$ The Punch: $60$ customers $\newline$ The Herald: $50$ customers
Identify Customers with Two Papers: Identify the number of customers buying two papers. $\newline$ The Times and The Punch: $17$ customers $\newline$ The Punch and The Herald: $15$ customers $\newline$ The Herald and The Times: $16$ customers
Identify Customers with All Three Papers: Identify the number of customers buying all three papers. All three papers: $3$ customers
Use Inclusion-Exclusion Principle: Use the principle of inclusion-exclusion to find the total number of unique customers. $\newline$ Total unique customers = (Customers buying The Times) + (Customers buying The Punch) + (Customers buying The Herald) - (Customers buying The Times and The Punch) - (Customers buying The Punch and The Herald) - (Customers buying The Herald and The Times) + (Customers buying all three papers) $\newline$ Total unique customers = $70 + 60 + 50 - 17 - 15 - 16 + 3$
Perform Calculation: Perform the calculation from Step $4$ . $\newline$ Total unique customers = $70 + 60 + 50 - 17 - 15 - 16 + 3$ $\newline$ Total unique customers = $180 - 48 + 3$ $\newline$ Total unique customers = $132 + 3$ $\newline$ Total unique customers = $135$