1. **State the problem:** We have 200 tennis enthusiasts surveyed about attendance at three grand slam tournaments: Australian Open (A), Wimbledon (B), and US Open (C). Given attendance numbers and overlaps, we want to find the number of people in each region of the Venn diagram. 2. **Given data:** - Total surveyed: $200$ - $|A|=65$ - $|B|=100$ - $|C|=110$ - $|A \cap B|=25$ - $|A \cap C|=30$ - $|B \cap C|=40$ - $|A \cap B \cap C|=15$ 3. **Define variables for each region:** - $a$: only Australian Open - $d$: Australian Open and Wimbledon only - $b$: Australian Open and US Open only - $e$: all three tournaments - $g$: only Wimbledon - $f$: Wimbledon and US Open only - $c$: only US Open - $h$: none of the three 4. **Use the triple intersection to find overlaps only:** - $e = 15$ - $d = |A \cap B| - e = 25 - 15 = 10$ - $b = |A \cap C| - e = 30 - 15 = 15$ - $f = |B \cap C| - e = 40 - 15 = 25$ 5. **Find only Australian Open attendees:** $$a = |A| - (d + b + e) = 65 - (10 + 15 + 15) = 65 - 40 = 25$$ 6. **Find only Wimbledon attendees:** $$g = |B| - (d + f + e) = 100 - (10 + 25 + 15) = 100 - 50 = 50$$ 7. **Find only US Open attendees:** $$c = |C| - (b + f + e) = 110 - (15 + 25 + 15) = 110 - 55 = 55$$ 8. **Find those who attended none:** $$h = 200 - (a + d + b + e + g + f + c) = 200 - (25 + 10 + 15 + 15 + 50 + 25 + 55) = 200 - 195 = 5$$ 9. **Summary of values:** - $a=25$ - $d=10$ - $b=15$ - $e=15$ - $g=50$ - $f=25$ - $c=55$ - $h=5$ These values satisfy all given conditions and complete the Venn diagram. **Final answer:** $$a=25, d=10, b=15, e=15, g=50, f=25, c=55, h=5$$