Question

in a race in which eight automobiles are entered and there are no ties, in how many ways can the first three finishers come in?
ways

Explanation:

Step1: Determine the formula

This is a permutation problem. The formula for permutations is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be selected and arranged. Here, \(n = 8\) (total number of automobiles) and \(r=3\) (number of finishers we are interested in).

Step2: Calculate the factorial values

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So, \(8! = 8\times7\times6\times5\times4\times3\times2\times1\) and \((8 - 3)!=5!=5\times4\times3\times2\times1\). Then \(P(8,3)=\frac{8!}{5!}=\frac{8\times7\times6\times5!}{5!}\).

Step3: Simplify the expression

Cancel out the \(5!\) terms. We get \(P(8,3)=8\times7\times6\).

$$8\times7\times6=336$$

Answer:

\(336\)