2020 AMC 8 Problems/Problem 23

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

Credits to asbodke for the solution: Without the restriction that each student receives one award, the answer is $3^5=243$ . We then subtract the cases where one student doesn't receive an award. There are 3 students to choose from, and then after that, there are $2^5=32$ options, so we subtract $3\cdot32=96$ options. However, the cases where one student receives the awards were counted once originally, but then subtracted twice! However, we only want them to be counted zero times, so we have to add them back again. There are 3 cases when this happens (since there are 3 students). Thus, the answer is $243-96+3=150\implies\boxed{\textbf{(B)}150}$ .

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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2020 AMC 8 Problems/Problem 23

Solution 1

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