How many ways can 3 identical prizes be awarded to 98 potential winners?

In summary, there are 98 ways to give the first prize, 97 ways to give the second prize, and 96 ways to give the third prize, for a total of (98)(97)(96) possibilities. However, because the prizes are identical, we must divide by the number of ways to order 3 people, which is 3!. This gives us a final answer of (98)(97)(96)/3! = 156,078 possible ways to award the prizes.
  • #1
theRukus
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Homework Statement


How many was can 3 identical prizes be awarded to 98 potential winners?


Homework Equations





The Attempt at a Solution


Well. I know that if the prizes were unique, the first prize would have 98 possible winners, the second prize would have 97 possible winners, and the third prize would have 96 possible winners, totaling (98)(97)(96) possibilities. Since they are all identical, some number of solutions will be eliminated. Can anyone hint at how I should go about this?
 
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  • #2
can one winner have more that one prize?
 
  • #3
No, they're restricted to one prize each.
 
  • #4
Yes, you are right that there are 98 ways to give the first prize to someone, 97 ways to give the second prize, and 96 ways to give the third. So if there were three different prizes, there would be 98(97)(96) ways to give out the three prizes.
(Which is equal, by the way, to
[tex]\frac{98!}{95!}= \frac{98!}{(98- 3)!}[/tex]

But because the three prized are the same, it does not matter in which order we pick the three people. How many different orders are there for 3 people? So how many times is each set of 3 people being counted? Divide by that.

Does that answer look familiar? Here's another way to do the same problem: Label each person who is to be given a prize with an "A", label each person who is not with a "B". Different choices of people can be thought of as different orders for the "A"s and "B"s. How many different ways can you order 3 "A"s and 95 "B"s?
 
Question 1: What is a "discrete counting question"?

A discrete counting question is a type of mathematical problem that involves counting a finite number of distinct objects or events. It often involves using combinations or permutations to determine the total number of possible outcomes.

Question 2: What is the difference between a discrete counting question and a continuous counting question?

The main difference between a discrete counting question and a continuous counting question is that discrete counting involves counting a finite number of distinct objects or events, while continuous counting involves measuring a continuous range of values, such as time, length, or weight.

Question 3: How do I solve a discrete counting question?

To solve a discrete counting question, you first need to identify the type of problem you are dealing with, such as a combination or permutation problem. Then, you can use the appropriate formula and plug in the given values to calculate the total number of outcomes. It is important to carefully read and understand the question to ensure you are using the correct formula.

Question 4: What is the purpose of discrete counting in science?

Discrete counting is used in science to determine the total number of possible outcomes in a given situation. This can be useful in genetics, probability and statistics, and other areas of science where the number of possible outcomes needs to be known.

Question 5: Can discrete counting be applied in real-world situations?

Yes, discrete counting can be applied in real-world situations. For example, it can be used in genetics to calculate the probability of certain traits appearing in offspring, or in analyzing data sets to determine the probability of certain outcomes. It is a useful tool in many scientific fields where counting and probability are involved.

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