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 = 120B.Combination; number of ways = 120C.Permutation; number of ways = 1D.Combination; number of ways = 1
Question
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 = 120B.Combination; number of ways = 120C.Permutation; number of ways = 1D.Combination; number of ways = 1
Solution
This situation involves a permutation. A permutation is used when the order of selection matters, which is the case here because each medal represents a different place in the race (first, second, third, fourth, fifth).
The formula for permutations is nPr = n! / (n - r)!, where n is the total number of options, r is the number of options chosen from the total, and "!" denotes a factorial, which means multiplying a number by every number below it down to 1.
In this case, we have 5 runners and 5 medals, so n = 5 and r = 5.
So, the number of ways to award the medals is 5P5 = 5! / (5 - 5)! = 5! / 0!
Since 0! is defined as 1, this simplifies to 5!, which is 5 * 4 * 3 * 2 * 1 = 120.
So, the answer is A. Permutation; number of ways = 120.
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