1. **State the problem:** We have 460 people surveyed. 100 read no newspaper, so 360 read at least one of the three newspapers: Daily Times (DT), Guidance (G), and Punch (P). Given: - Total readers $= 360$ - $|DT|=102$, $|G|=141$, $|P|=120$ - $|DT \cap G|=74$, $|DT \cap P|=62$, $|G \cap P|=35$ Find: (a) Number who read all three newspapers $|DT \cap G \cap P|$ (b) Number who read exactly two newspapers (c) Number who read exactly one newspaper (d) Number who read Punch only 2. **Formula and rules:** Use the principle of inclusion-exclusion for three sets: $$|DT \cup G \cup P| = |DT| + |G| + |P| - |DT \cap G| - |DT \cap P| - |G \cap P| + |DT \cap G \cap P|$$ Since all 360 readers read at least one newspaper, $|DT \cup G \cup P|=360$. 3. **Find $|DT \cap G \cap P|$:** $$360 = 102 + 141 + 120 - 74 - 62 - 35 + |DT \cap G \cap P|$$ Calculate the sum: $$360 = 363 - 171 + |DT \cap G \cap P|$$ $$360 = 192 + |DT \cap G \cap P|$$ $$|DT \cap G \cap P| = 360 - 192 = 168$$ 4. **Find exactly two newspapers:** Number reading exactly two = sum of pairwise intersections minus thrice the triple intersection (since triple counted in all pairs): $$= (|DT \cap G| + |DT \cap P| + |G \cap P|) - 3 \times |DT \cap G \cap P|$$ $$= (74 + 62 + 35) - 3 \times 168 = 171 - 504 = -333$$ This negative value is impossible, indicating an error in the triple intersection calculation. Re-examine step 3: $$360 = 102 + 141 + 120 - 74 - 62 - 35 + x$$ $$360 = 363 - 171 + x$$ $$360 = 192 + x$$ $$x = 168$$ Since $x=168$ is greater than any pairwise intersection, this is impossible. 5. **Correct approach:** The triple intersection cannot exceed any pairwise intersection. Let's denote $x = |DT \cap G \cap P|$. Rewrite inclusion-exclusion: $$360 = 102 + 141 + 120 - 74 - 62 - 35 + x$$ $$360 = 363 - 171 + x$$ $$360 = 192 + x$$ $$x = 168$$ Since $x$ cannot be 168, the problem data is inconsistent or misinterpreted. 6. **Assuming the problem means the number who read all three is $x$, and the pairwise intersections include the triple intersection, then:** Number who read exactly two newspapers: $$= (|DT \cap G| - x) + (|DT \cap P| - x) + (|G \cap P| - x) = (74 - x) + (62 - x) + (35 - x) = 171 - 3x$$ Number who read exactly one newspaper: $$= |DT| + |G| + |P| - 2 \times \text{(exactly two)} - 3x$$ But better to use: $$\text{Exactly one} = |DT| - (|DT \cap G| + |DT \cap P| - x) + |G| - (|DT \cap G| + |G \cap P| - x) + |P| - (|DT \cap P| + |G \cap P| - x)$$ Simplify: $$= (102 - (74 + 62 - x)) + (141 - (74 + 35 - x)) + (120 - (62 + 35 - x))$$ $$= (102 - 136 + x) + (141 - 109 + x) + (120 - 97 + x)$$ $$= (-34 + x) + (32 + x) + (23 + x) = 21 + 3x$$ 7. **Use total readers:** $$360 = \text{exactly one} + \text{exactly two} + \text{all three}$$ $$360 = (21 + 3x) + (171 - 3x) + x = 21 + 3x + 171 - 3x + x = 192 + x$$ $$x = 360 - 192 = 168$$ Again, $x=168$ is impossible. 8. **Conclusion:** The data is inconsistent as given. However, if we assume the triple intersection is $x$, then: - $x = 168$ (impossible) - Number reading exactly two = $171 - 3x$ - Number reading exactly one = $21 + 3x$ - Number reading Punch only = $|P| - (|DT \cap P| + |G \cap P| - x) = 120 - (62 + 35 - x) = 120 - 97 + x = 23 + x$ Since $x=168$ is impossible, the problem data likely contains an error. **Final answers assuming data is consistent and $x=168$ (for demonstration):** (a) $|DT \cap G \cap P| = 168$ (b) Exactly two newspapers $= 171 - 3 \times 168 = -333$ (impossible) (c) Exactly one newspaper $= 21 + 3 \times 168 = 525$ (d) Punch only $= 23 + 168 = 191$ **Note:** The negative and large values indicate inconsistency in the problem data.