Problem:
In a race among 5 snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
Answer Choices:
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Solution:
If there are no ties, then there are 5! possible race results. Suppose of the 5 snails are tied, where . There are ways to choose the snails that are tied, and then considering those snails as a group, there are entrants and therefore ! orders of finish. The number of possible results is thus
The problems on this page are the property of the MAA's American Mathematics Competitions