AMC 10 Weekly Practice Round 3
Complete problem set with solutions and individual problem pages
Five students and participate in a certain technical competition and receive the first through fifth places. and ask about the results. The organizer tells : “Unfortunately, neither you nor won the championship.” To , the organizer says: “Of course, you are not in the last place.” Based on the organizer’s responses, how many different possible rankings of the five students are there?
- A.
- B.
- C.
- D.
- E.
According to the problem, and did not win the championship, and is not in the last place. We discuss two cases:
1. If is in the last place, then can be in second, third, or fourth place. Thus, there are possible positions for , and the remaining three students have possible arrangements. In this case, there are
possible rankings.
2. If is not in the last place, then and must be placed in the second, third, and fourth positions. There are possible arrangements for and , and the remaining three students also have possible arrangements. In this case, there are
possible rankings.
Therefore, in total there are
different possible rankings.
Hence, the answer is .
