Chapter 0.4, Problem 13E

To determine

To find:The number of ways to choose the winner and first, second and third runners-up in a contest.

Answer to Problem 13E

There are 5040 ways to choose the winner and first, second and third runners-up in a contest with 10 finalists.

Explanation of Solution

Given information:There are 10 finalists.

Calculation:Since the order in which the finalists are arranged determines the ranking of the winner, and first, second and third runners-up, this situation involves permutations.

  $\begin{array}{l}P{}_4\\ =\frac{10!}{(10-4)!}\\ =\frac{10!}{6!}\\ =\frac{10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =10\cdot 9\cdot 8\cdot 7\\ =5040\end{array}$

Thus, there are 5040 ways to choose the winner and first, second and third runners-up in a contest with 10 finalists.