1. **Problem statement:** We have three magazines: Mwananchi (M), Nipashe (N), and Daily News (D). Given: - $|M|=70$, $|N|=60$, $|D|=50$ - $|M \cap N|=17$, $|N \cap D|=15$, $|D \cap M|=16$ We need to find: - Number of customers who bought only Mwananchi and Nipashe - Number who bought only Nipashe and Daily News - Number who bought only Daily News and Mwananchi - Number who bought only Nipashe - Number who bought only Mwananchi - Number who bought only Daily News - Total number of customers 2. **Important rules and formula:** - Use the principle of inclusion-exclusion for three sets: $$|M \cup N \cup D| = |M| + |N| + |D| - |M \cap N| - |N \cap D| - |D \cap M| + |M \cap N \cap D|$$ - Let $x$ be the number of customers who bought all three magazines: $|M \cap N \cap D| = x$ 3. **Define variables for only two sets:** - Only Mwananchi and Nipashe: $a = |M \cap N| - x = 17 - x$ - Only Nipashe and Daily News: $b = |N \cap D| - x = 15 - x$ - Only Daily News and Mwananchi: $c = |D \cap M| - x = 16 - x$ 4. **Define variables for only one set:** - Only Mwananchi: $m = |M| - a - c - x = 70 - (17 - x) - (16 - x) - x = 70 - 17 - 16 + x + x - x = 37 + x$ - Only Nipashe: $n = |N| - a - b - x = 60 - (17 - x) - (15 - x) - x = 60 - 17 - 15 + x + x - x = 28 + x$ - Only Daily News: $d = |D| - b - c - x = 50 - (15 - x) - (16 - x) - x = 50 - 15 - 16 + x + x - x = 19 + x$ 5. **Total customers:** $$T = m + n + d + a + b + c + x$$ Substitute: $$T = (37 + x) + (28 + x) + (19 + x) + (17 - x) + (15 - x) + (16 - x) + x$$ Simplify: $$T = 37 + 28 + 19 + 17 + 15 + 16 + x + x + x - x - x - x + x = 132 + x$$ 6. **Assuming no customers bought all three (i.e., $x=0$) since not given:** - Only Mwananchi and Nipashe: $a = 17$ - Only Nipashe and Daily News: $b = 15$ - Only Daily News and Mwananchi: $c = 16$ - Only Mwananchi: $m = 37$ - Only Nipashe: $n = 28$ - Only Daily News: $d = 19$ - Total customers: $T = 132$ **Final answers:** - Mwananchi and Nipashe only: 17 - Nipashe and Daily News only: 15 - Daily News and Mwananchi only: 16 - Nipashe only: 28 - Mwananchi only: 37 - Daily News only: 19 - Total customers: 132 This completes the solution.