At a competition with 5 runners, medals are awarded for first, second, and third places. Each of the 3 medals is different. How many ways are there to award the medals?
Decide if this is a permutation or a combination, and find the number of ways to award the medals.
A. Permutation; number of ways = 10
B. Combination; number of ways = 60
C. Combination, number of ways = 10
D. Permutation; number of ways = 60
Answer (2)
Recognize the problem as a permutation since the order of awarding medals matters.
Apply the permutation formula: P ( n , k ) = ( n − k )! n ! .
Substitute n = 5 and k = 3 into the formula: P ( 5 , 3 ) = 2 ! 5 ! = 2 120 .
Calculate the result: 60 .
Explanation
Analyze the problem We have 5 runners, and we want to award 3 different medals (first, second, and third place). Since the order of the runners matters (getting first place is different from getting second place), this is a permutation problem.
State the permutation formula The formula for permutations is given by: P ( n , k ) = ( n − k )! n ! where n is the total number of items, and k is the number of items we want to choose. In our case, n = 5 (the total number of runners) and k = 3 (the number of medals to award).
Calculate the number of permutations Now, we plug in the values of n and k into the formula: P ( 5 , 3 ) = ( 5 − 3 )! 5 ! = 2 ! 5 ! We calculate the factorials: 5 ! = 5 × 4 × 3 × 2 × 1 = 120 2 ! = 2 × 1 = 2 So, we have: P ( 5 , 3 ) = 2 120 = 60
State the final answer Therefore, there are 60 ways to award the medals. This corresponds to option D.
Examples
Consider a scenario where a coach needs to select a batting order for the first three batters out of a team of five players. Since the order in which the players bat matters, this is a permutation problem. The coach needs to determine the number of different batting orders possible for those first three slots. This is directly analogous to awarding medals to runners, where each position (1st, 2nd, 3rd) is distinct and the order matters.
The number of ways to award medals to 5 runners is calculated using permutations, as the order of medals matters. By applying the permutation formula, we find that there are 60 different ways to award the medals. Therefore, the correct option is D.
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