Question

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

the first three finishers can finish in \\(\square\\) ways.

Explanation:

Step1: Use permutation formula

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 5\) (total number of automobiles) and \(r=3\) (number of positions - first, second, third).

Step2: Calculate factorial values

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So \(P(5,3)=\frac{5!}{(5 - 3)!}=\frac{5!}{2!}\).
Since \(5!=5\times4\times3\times2\times1 = 120\) and \(2!=2\times1=2\).

Step3: Compute the result

\(P(5,3)=\frac{120}{2}=60\).

Answer:

\(60\)