1. **Problem Statement:** We analyze the sets of children who played Fortnite (F), Minecraft (M), and League of Legends (L) based on the given data. 2. **Given Data:** - $|F|=95$ - $|M \text{ only}|=34$ - $|F \cap M \cap L|=15$ - $|\text{at least two games}|=50$ - $|M \cap L|=22$ - $|F \cap L \text{ but not } M|=18$ - $|\text{neither } F \text{ nor } M|=64$ - $|L \cup M|=104$ 3. **Goal:** Find the number of children in each section of the Venn diagram and explain each. 4. **Step 1: Define variables for unknowns:** - Let $x = |F \cap M \text{ only}|$ - Let $y = |L \text{ only}|$ - Let $z = |F \text{ only}|$ 5. **Step 2: Use given intersections:** - $|F \cap M \cap L|=15$ - $|F \cap L \text{ but not } M|=18$ - $|M \cap L|=22$ includes $|F \cap M \cap L|=15$, so $|M \cap L \text{ only}|=22-15=7$ 6. **Step 3: Use "at least two games" count:** - $|\text{at least two}|=50 = |F \cap M \text{ only}| + |F \cap L \text{ only}| + |M \cap L \text{ only}| + |F \cap M \cap L|$ - Substitute known values: $50 = x + 18 + 7 + 15$ - Simplify: $x + 40 = 50 \Rightarrow x = 10$ 7. **Step 4: Use $|M \text{ only}|=34$ and $|M|$ total:** - $|M| = |M \text{ only}| + |F \cap M \text{ only}| + |M \cap L \text{ only}| + |F \cap M \cap L|$ - $|M| = 34 + 10 + 7 + 15 = 66$ 8. **Step 5: Use $|L \cup M|=104$ to find $|L|$:** - $|L \cup M| = |L| + |M| - |L \cap M|$ - $104 = |L| + 66 - 22$ - $|L| = 104 - 66 + 22 = 60$ 9. **Step 6: Find $|L \text{ only}|=y$:** - $|L| = |L \text{ only}| + |F \cap L \text{ only}| + |M \cap L \text{ only}| + |F \cap M \cap L|$ - $60 = y + 18 + 7 + 15$ - $y + 40 = 60 \Rightarrow y = 20$ 10. **Step 7: Find $|F \text{ only}|=z$ using $|F|=95$:** - $|F| = |F \text{ only}| + |F \cap M \text{ only}| + |F \cap L \text{ only}| + |F \cap M \cap L|$ - $95 = z + 10 + 18 + 15$ - $z + 43 = 95 \Rightarrow z = 52$ 11. **Step 8: Verify total children:** - Total playing at least one game = sum of all disjoint sections: $z + 34 + x + y + 7 + 18 + 15 = 52 + 34 + 10 + 20 + 7 + 18 + 15 = 156$ - Add neither $F$ nor $M$ (64) to get total children: $156 + 64 = 220$ 12. **Explanation of each section:** - $F \text{ only} = 52$: Played only Fortnite. - $M \text{ only} = 34$: Played only Minecraft. - $L \text{ only} = 20$: Played only League of Legends. - $F \cap M \text{ only} = 10$: Played Fortnite and Minecraft but not League. - $F \cap L \text{ only} = 18$: Played Fortnite and League but not Minecraft. - $M \cap L \text{ only} = 7$: Played Minecraft and League but not Fortnite. - $F \cap M \cap L = 15$: Played all three games. - Neither $F$ nor $M = 64$: Played none of Fortnite or Minecraft (may or may not have played League). **Final answer:** The Venn diagram sections are $52, 34, 20, 10, 18, 7, 15$ for the respective regions described above.