2024 AMC 10B Problem 17
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?
If there is no tie, then there are
possible results.
possible results.
If there is one tie, organize the casework by the number of snails involved in the tie.
Case 1: All 5 snails are tied.
There is exactly one possible result.
There is exactly one possible result.
Case 2: Exactly 4 snails are tied.
There are ways to select the 4 tied snails. The tied group can finish either first or second, and the remaining snail occupies the other position.
There are ways to select the 4 tied snails. The tied group can finish either first or second, and the remaining snail occupies the other position.
Case 3: Exactly 3 snails are tied.
Choose the 3 tied snails, then arrange the tied group and the other 2 snails in the 3 available positions.
Choose the 3 tied snails, then arrange the tied group and the other 2 snails in the 3 available positions.
Case 4: Exactly 2 snails are tied.
Choose the 2 tied snails, then arrange the tied group and the other 3 snails in the 4 available positions.
Choose the 2 tied snails, then arrange the tied group and the other 3 snails in the 4 available positions.
By Hints 1, 3, 4, 5, and 6, the total number of possible results is
(D) 431
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