1. **Problem Statement:** Seven committees must elect a chairperson to represent them at the end-of-year board meeting. Some people serve on more than one committee and cannot be elected chairperson for more than one committee. We need to find a system of distinct representatives (SDR) where each committee is assigned a unique chairperson from its members. 2. **Understanding the Problem:** This is a classic problem of finding a system of distinct representatives, which can be modeled as finding a matching in a bipartite graph. One set of vertices represents the committees, and the other set represents the members. An edge connects a committee to a member if that member belongs to the committee. 3. **Modeling the Problem:** Committees: Benefits (Bf), Computing (C), Purchasing (P), Recruitment (R), Refreshments (Rf), Social Media (SM), Travel Expenses (TE) Members: Agatha (A), Dinah (D), Evan (E), Vlad (V), Nancy (N), Leah (L), Omar (O), George (G) Edges: - Bf: A, D, E, V - C: E, N, L, O - P: G, V, L - R: D, O, A - Rf: N, G - SM: E, L, V, O - TE: A, V, G 4. **Goal:** Find a matching where each committee is matched to a unique member who is on that committee, and no member is assigned to more than one committee. 5. **Step-by-step Matching:** - Assign Refreshments (Rf) to Nancy (N) because N is only in Rf and C (but C has other options). - Assign Purchasing (P) to George (G) because G is in P, Rf (assigned), and TE. - Assign Travel Expenses (TE) to Vlad (V) because V is in Bf, P (assigned), SM, TE. - Assign Benefits (Bf) to Evan (E) because E is in Bf, C, SM. - Assign Computing (C) to Leah (L) because L is in C, P (assigned), SM. - Assign Recruitment (R) to Dinah (D) because D is in Bf (assigned), R. - Assign Social Media (SM) to Omar (O) because O is in C (assigned), R (assigned), SM. 6. **Verification:** Each committee has a unique member assigned: - Bf: E - C: L - P: G - R: D - Rf: N - SM: O - TE: V No member is assigned to more than one committee. 7. **Conclusion:** A system of distinct representatives exists and is: Benefits - Evan Computing - Leah Purchasing - George Recruitment - Dinah Refreshments - Nancy Social Media - Omar Travel Expenses - Vlad This ensures each committee has a unique chairperson who is a member of that committee.