Permutations and Combinations questions

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Five separate awards are to be presented to select students from a class of 30. How many different outcomes are possible if:

1. If a student can receive any number of awards

2. If each student can receive at most one award

I tried like $30!/5!(30-5)!$ but seens not working

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2 Answers

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For the first problem (any number of awards), line up the $5$ trophies. There are $30$ choices for who receives the first trophy. For each such choice, there are $30$ choices for who receives the second. So there are $30^2$ ways to award the first two trophies. For each such way, there are $30$ ways to award the third trophy, and so on, for a total of $30^5$.

For the second problem, the reasoning is almost the same. There are $30$ ways to award the first trophy. For each such way there are $29$ ways to award the second trophy, and so on for a total of $(30)(29)(28)(27)(26)$.

There are other approaches to the second problem. For example, we can choose the $5$ people who will get trophies in $\binom{30}{5}$ ways. For each such choice, the decision as to who gets what can be done in $(5)(4)(3)(2)(1)=5!$ ways, for a total of $\binom{30}{5}5!$. This expression, when evaluated, yields $(30)(29)(28)(27)(26)$.

Remark: In order to get the right answer, it can be very useful to visualize the awards process in a very concrete way. So for the first problem, think of five medals, gold, silver, bronze, plastic, cardboard. The cardboard medal can be awarded in $30$ ways. For each such way, the plastic medal can be awarded in $30$ ways, and so on.

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If you can notice for the first part, you can give your award to 30 students. Now one of them only can get the next award out of 30 again. So the second might go to the same guy, or the next one, and so on which makes 30 separate cases for every student. So total no. of cases is $ 30*30=900$. You could see the pattern again that for the third trophy you have 900 separate cases for 30 students, making the total to $30^3$ and if you continue for five trophies, the count scales to $30^5$.

Now for the second case, you can see again 30 cases for first trophy, but only 29 for the second (and so on till 26 for the 5th). So simply do $30*29*28*27*26$ to get the second part. Or simply $(30!)/(25!)$

Hope this helps!

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