1. **State the problem:** We have 100 voters choosing among three candidates: Akayuure (A), Manukre (M), and Odonti (O). We want to find the number of voters who preferred all three candidates (A \cap M \cap O) and determine if any candidate won unopposed by majority (more than 50 votes). 2. **Given data:** - Total voters: 100 - $|A \cap M| = 14$ - $|A \cup M \text{ but not } O| = 49$ - $|M \text{ but not } O \text{ or } A| = 21$ - $|M \cup O \text{ but not } A| = 61$ - $|O \text{ but not } A \text{ or } M| = 32$ - $|A \cap O \text{ but not } M| = 7$ - Every voter voted for at least one candidate. 3. **Define variables for the Venn diagram regions:** - Let $x = |A \cap M \cap O|$ (all three) - Let $a = |A \text{ only}|$ - Let $b = |M \text{ only}|$ - Let $c = |O \text{ only}|$ - Let $d = |A \cap M \text{ only}| = |A \cap M| - x = 14 - x$ - Let $e = |M \cap O \text{ only}| = ?$ - Let $f = |A \cap O \text{ only}| = 7$ 4. **Use given data to find $e$ and other values:** - $|A \cup M \text{ but not } O| = a + b + d = 49$ - $|M \text{ but not } O \text{ or } A| = b = 21$ - $|M \cup O \text{ but not } A| = b + c + e = 61$ - $|O \text{ but not } A \text{ or } M| = c = 32$ From $b = 21$ and $c = 32$, substitute into $b + c + e = 61$: $$21 + 32 + e = 61 \implies e = 61 - 53 = 8$$ From $a + b + d = 49$ and $b = 21$, $d = 14 - x$: $$a + 21 + (14 - x) = 49 \implies a = 49 - 21 - 14 + x = 14 + x$$ 5. **Sum all regions to total 100:** $$a + b + c + d + e + f + x = 100$$ Substitute known values: $$(14 + x) + 21 + 32 + (14 - x) + 8 + 7 + x = 100$$ Simplify: $$14 + x + 21 + 32 + 14 - x + 8 + 7 + x = 100$$ $$14 + 21 + 32 + 14 + 8 + 7 + x = 100$$ $$96 + x = 100 \implies x = 4$$ 6. **Calculate each region:** - $x = 4$ - $a = 14 + 4 = 18$ - $b = 21$ - $c = 32$ - $d = 14 - 4 = 10$ - $e = 8$ - $f = 7$ 7. **Determine total votes per candidate:** - Akayuure: $a + d + f + x = 18 + 10 + 7 + 4 = 39$ - Manukre: $b + d + e + x = 21 + 10 + 8 + 4 = 43$ - Odonti: $c + e + f + x = 32 + 8 + 7 + 4 = 51$ 8. **Conclusion:** - Number of voters favoring all three candidates is $\boxed{4}$. - Odonti received 51 votes, which is more than 50% of 100, so Odonti won unopposed by majority.