At a competition with 5 runners, 5 medals are awarded for first place through fifth place. Each medal is different. How many ways are there to award the medals?
Decide if the situation involves a permutation or a combination, and then find the number of ways to award the medals.
A. Permutation; number of ways = 120
B. Combination; number of ways = 120
C. Permutation; number of ways = 1
D. Combination; number of ways = 1
Answer (1)
Determine that the problem is a permutation since the order of the runners matters.
Apply the permutation formula: P ( 5 , 5 ) = ( 5 − 5 )! 5 ! .
Simplify the formula to 5 ! = 5 × 4 × 3 × 2 × 1 .
Calculate the result: 5 ! = 120 , so there are 120 ways to award the medals.
Explanation
Identify the problem type We have 5 runners and 5 distinct medals to award. The order in which the runners finish matters because each medal is different (1st place, 2nd place, etc.). Therefore, this is a permutation problem.
State the permutation formula To find the number of ways to award the medals, we need to calculate the number of permutations of 5 runners taken 5 at a time. The formula for permutations is given by: P ( n , r ) = ( n − r )! n ! In this case, n = 5 (number of runners) and r = 5 (number of medals).
Apply the formula Plugging in the values, we get: P ( 5 , 5 ) = ( 5 − 5 )! 5 ! = 0 ! 5 ! = 5 ! Since 0 ! = 1 , we have P ( 5 , 5 ) = 5 ! .
Calculate the result Now, we calculate 5 ! :
5 ! = 5 × 4 × 3 × 2 × 1 = 120 So, there are 120 ways to award the medals.
Final Answer The situation involves a permutation, and the number of ways to award the medals is 120. Therefore, the correct answer is A.
Examples
Consider a scenario where you are arranging books on a shelf. If you have 5 different books and want to arrange all of them, the number of ways you can do this is a permutation problem. This is because the order in which you place the books matters. The calculation of permutations helps in determining the number of possible arrangements, which can be useful in various scenarios such as scheduling events, arranging items, or even in cryptography.
