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Chapter 11: Problem 69

Six horses are entered in a race. If two horses are tied for first place, and there are no ties among the other four horses, in how many ways can the six horses cross the finish line?

Short Answer

Expert verified
The six horses can cross the finish line in 2880 different ways.

Step by step solution

01

Identify the number of permutations for the first set of horses

The two horses tied for the first place can be considered as a single entity. So including this entity, we effectively have five entities to arrange. The number of permutations of n distinct entities is given by the formula \(n!\), where ! denotes factorial. So the number of permutations of our five entities is \(5! = 5 * 4 * 3 * 2 * 1 = 120\).
02

Permutations for the second set of horses

The four remaining horses can finish in \(4!\) ways. Hence, number of ways the four remaining horses can finish = \(4 * 3 * 2 * 1 = 24\).
03

Total number of ways all horses can finish

The total number of ways the horses can finish is obtained by multiplying the permutations found in step 1 and step 2. Therefore, total number of ways = \(120 * 24 = 2880\)

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