(a) 56 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?(b) An experiment involves 31 participants. From these, a group of 4 participants is to be tested under a special condition. How many groups of 4 participants are possible?
Question
(a) 56 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed?(b) An experiment involves 31 participants. From these, a group of 4 participants is to be tested under a special condition. How many groups of 4 participants are possible?
Solution
(a) The number of ways the 3 medals can be distributed is given by the permutation of 56 athletes taken 3 at a time. This is calculated as 56P3 = 56! / (56-3)!.
Step 1: Calculate 56 factorial (56!) which is 56555453...321. Step 2: Calculate (56-3) factorial which is 53 factorial (53!) which is 535251...32*1. Step 3: Divide 56! by 53! to get the number of ways the medals can be distributed.
(b) The number of groups of 4 participants that can be chosen from 31 is given by the combination of 31 participants taken 4 at a time. This is calculated as 31C4 = 31! / [4!(31-4)!].
Step 1: Calculate 31 factorial (31!) which is 313029*...321. Step 2: Calculate 4 factorial (4!) which is 4321. Step 3: Calculate (31-4) factorial which is 27 factorial (27!) which is 272625*...32*1. Step 4: Divide 31! by the product of 4! and 27! to get the number of possible groups.
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