A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a c

Question

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. In how many ways can this be done if either Alice or Ben must be chairperson?

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1 (Break into Cases):","type":"text"},{"text":" We want to select a chairperson, a secretary, and a treasurer from 6 people under the condition that either Alice or Ben must be the chairperson. The choices are sequential and without replacement, so we analyze two disjoint cases:","type":"text"}],"type":"paragraph"},{"content":[{"content":[{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Case 1: Alice is chairperson."}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Chairperson slot: 1 choice (Alice)","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"type":"text","text":"Secretary slot: 5 remaining choices"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Treasurer slot: 4 remaining choices","type":"text"}],"type":"paragraph"}],"type":"list_item"}],"type":"bullet_list"},{"type":"paragraph","content":[{"type":"text","text":"  Total for Case 1 = "},{"attrs":{"latex":"1 \times 5 \times 4 = 20"},"type":"math_inline"},{"text":" ways.","type":"text"}]},{"type":"bullet_list","content":[{"content":[{"content":[{"text":"Case 2: Ben is chairperson.","type":"text","marks":[{"type":"bold"}]}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Chairperson slot: 1 choice (Ben)","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"content":[{"text":"Secretary slot: 5 remaining choices","type":"text"}],"type":"paragraph"}],"type":"list_item"},{"content":[{"type":"paragraph","content":[{"text":"Treasurer slot: 4 remaining choices","type":"text"}]}],"type":"list_item"}]},{"content":[{"text":"  Total for Case 2 = ","type":"text"},{"attrs":{"latex":"1 \times 5 \times 4 = 20"},"type":"math_inline"},{"text":" ways.","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 2 (Sum the Cases):"},{"type":"text","text":" Since the cases are disjoint, we sum their possibilities:"}],"type":"paragraph"},{"attrs":{"latex":"\text{Total Ways} = 20 \text{ (Case 1)} + 20 \text{ (Case 2)} = 40"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3 (Conclusion):","type":"text"},{"text":" There are exactly ","type":"text"},{"marks":[{"type":"code"}],"text":"40","type":"text"},{"text":" distinct ways to form the committee.","type":"text"}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

Solution