1. **State the problem:** We have 240 students total. Among them, 176 are on the honor roll (set $H$), 48 are members of the varsity team (set $V$), and 36 are in both groups (set $B = H \cap V$). We want to find how many students are in neither group (set $N$). 2. **Formula used:** The principle of inclusion-exclusion for two sets states: $$|H \cup V| = |H| + |V| - |H \cap V|$$ where $|H \cup V|$ is the number of students in honor roll or varsity or both. 3. **Calculate the number in $H \cup V$:** $$|H \cup V| = 176 + 48 - 36 = 224 - 36 = 188$$ 4. **Calculate the number in neither group $N$:** Since total students are 240, $$|N| = 240 - |H \cup V| = 240 - 188 = 52$$ 5. **Summary:** - Students on honor roll only: $|H| - |B| = 176 - 36 = 140$ - Students on varsity only: $|V| - |B| = 48 - 36 = 12$ - Students in both: $36$ - Students in neither: $52$ This matches the Venn diagram areas: green only (140), blue only (12), yellow overlap (36), pink outside (52).