2020 AMC 8 Problems/Problem 23

Problem

Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?

$\textbf{(A) }120 \qquad \textbf{(B) }150 \qquad \textbf{(C) }180 \qquad \textbf{(D) }210 \qquad \textbf{(E) }240$

Solution 1 (complementary counting)

Without the restriction that each student receives at least one award, we could simply take each of the $5$ awards and choose one of the $3$ students to give it to, so that there would be $3^5=243$ ways to distribute the awards. We now need to subtract the cases where at least one student doesn't receive an award. If a student doesn't receive an award, there are $3$ choices for which student that is, then $2^5 = 32$ ways of choosing a student to receive each of the awards, for a total of $3 \cdot 32 = 96$ . However, if $2$ students both don't receive an award, then such a case would be counted twice among our $96$ , so we need to add back in these cases. Of course, $2$ students both not receiving an award is equivalent to only $1$ student receiving all $5$ awards, so there are simply $3$ choices for which student that would be. It follows that the total number of ways of distributing the awards is $243-96+3=\boxed{\textbf{(B) }150}$ .

This is confusing, so you do not need to know...

Solution 3 (variation of Solution 2)

If each student must receive at least one award, then, as in Solution 2, we deduce that the only possible ways to split up the $5$ awards are $3,1,1$ and $2,2,1$ (i.e. one student gets three awards and the others get one each, or two students each get two awards and the other student is left with the last one). In the first case, there are $3$ choices for which student gets $3$ awards, and $\binom{5}{3} = 10$ choices for which $3$ awards they get. We are then left with $2$ awards, and there are exactly $2$ choices depending on which remaining student gets which. This yields a total for this case of $3 \cdot 10 \cdot 2 = 60$ . For the second case, there are similarly $3$ choices for which student gets only $1$ award, and $5$ choices for which award he gets. There are then $4$ remaining awards, from which we choose $2$ to give to one student and $2$ to give to the other, which can be done in $\binom{4}{2} = 6$ ways (and we can say that e.g. the $2$ chosen this way go to the first remaining student and the other $2$ go to the second remaining student, which counts all possibilities). This means the total for the second case is $3 \cdot 5 \cdot 6 = 90$ , and the answer is $60 + 90 = \boxed{\textbf{(B) }150}$ .

Video Solution by WhyMath

https://youtu.be/HkFQe7ZxBb4

~savannahsolver

Video Solutions

https://youtu.be/tDChKU0pVN4
https://youtu.be/RUg6QfV5yg4

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=1443

~Interstigation

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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