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?
Solution 1
Firstly, observe that it is not possible for a single student to receive or
awards because this would mean that one of the other students receives no awards. Thus, each student must receive either
,
, or
awards. If a student receives
awards, then the other two students must each receive
award; if a student receives
awards, then another student must also receive
awards and the remaining student must receive
award. We consider each of these two cases in turn. If a student receives three awards, there are
ways to choose which student this is, and
ways to give that student
out of the
awards. Next, there are
students left and
awards to give out, with each student getting one award. There are clearly just
ways to distribute these two awards out, giving
ways to distribute the awards in this case.
In the other case, a student receives awards. We first have to choose which of the two students we will select to give two awards each to. There are
ways to do this, after which there are
ways to give the first student his two awards, leaving
awards yet to distribute. There are then \binom{3}{2}
2
1
1
1
\binom{3}{2}\cdot\binom{5}{2}\cdot\binom{3}{2}\cdot 1=90
60+90=\boxed{\textbf{(B) }150}$.
==Solution 2 (variation of Solution 1)==
If each student must receive at least one award, then, as in Solution 2, we deduce that the only possible ways to split up the$ (Error compiling LaTeX. Unknown error_msg)53,1,1
2,2,1
3
3
\binom{5}{3} = 10
3
2
2
3 \cdot 10 \cdot 2 = 60
3
1
5
4
2
2
\binom{4}{2} = 6
2
2
3 \cdot 5 \cdot 6 = 90
60 + 90 = \textbf{(B) }150$.
==Solution 3==
Without the restriction that each student receives at least one award, we could simply take each of the$ (Error compiling LaTeX. Unknown error_msg)53
3^5=243
3
2^5 = 32
3 \cdot 32 = 96
2
96
2
1
5
3
243-96+3=\boxed{\textbf{(B) }150}$.
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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