A newspaper agent sells equal number of three different newspapers P, Q and R to 327 persons.. 7 persons get Q and R, 9 get P and Q, 12 get P and R. 3 persons get all the three newspapers. How many persons get exactly one type newspaper?

Correct option is C

Given:

- Total number of persons: N = 327.
- Equal numbers of each newspaper are sold, so:
|$$P| = |Q| = |R| = \frac{327}{3} = 109$$​

Formula Used:

Categorize the persons based on the newspapers they buy:
- Exactly one type of newspaper.
- Exactly two types of newspapers.
- All three newspapers.

Solution:

1. Persons who get all three newspapers (P, Q, R):
- $$|P \cap Q \cap R| = 3$$​ .

2. Persons who get exactly two newspapers:
- Persons who get Q and R but not P:
​$$\text{Only } Q \text{ and } R = 7 - 3 = 4$$​
- Persons who get P and Q but not R:
​$$\text{Only } P \text{ and } Q = 9 - 3 = 6$$​
- Persons who get P and R but not Q:
​$$\text{Only } P \text{ and } R = 12 - 3 = 9$$​

3. Persons who get exactly one newspaper:
- For P:
​$$109 = \text{Exactly } P + \text{Only } P \text{ and } Q + \text{Only } P \text{ and } R + \text{All three}$$​
​$$\text{Exactly } P = 109 - (6 + 9 + 3) = 91$$​

- For Q:
​$$109 = \text{Exactly } Q + \text{Only } P \text{ and } Q + \text{Only } Q \text{ and } R + \text{All three}$$​
​$$\text{Exactly } Q = 109 - (6 + 4 + 3) = 96$$​

- For R:
​$$109 = \text{Exactly } R + \text{Only } Q \text{ and } R + \text{Only } P \text{ and } R + \text{All three}$$​
​$$\text{Exactly } R = 109 - (4 + 9 + 3) = 93$$​

4. Total persons who get exactly one newspaper:
​$$\text{Exactly one} = \text{Exactly } P + \text{Exactly } Q + \text{Exactly } R$$​
​$$\text{Exactly one} = 91 + 96 + 93 = 280$$​

Final Answer:

Option C: 280