Given that: In a race in which six automobiles are entered and there are no ties
To Determine: In how many ways can the first three finishers come in?
Solution:
This question can be solved using the permutations formula. If there are a total of n objects and r of these objects have to be ordered, the number of ways doing so is
[tex]^nP_r=\frac{n!}{(n-r)!}[/tex]The number of automobiles is 6. This is the number of objects out of 3 that have to be ordered. So, n=6 and r=3.
The number of ways in which the race can finish will be:
[tex]^6P_3=\frac{6!}{(6-3)!}[/tex][tex]\begin{gathered} ^6P_3=\frac{6!}{3!} \\ ^6P_3=\frac{6\times5\times4\times3!}{3!} \\ ^6P_3=\frac{6\times5\times4}{1} \\ ^6P_3=120 \end{gathered}[/tex]Hence, there are 120 ways for the first three finishers to come in
