2020 AMC 8 Problems/Problem 23
Contents
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?
Solution 1 (Principle of Inclusion-Exclusion)
Without the restriction that each student receives at least one award, we could simply take each of the awards and choose one of the
students to give it to, so that there would be
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
choices for which student that is, then
ways of choosing a student to receive each of the awards, for a total of
. However, if
students both don't receive an award, then such a case would be counted twice among our
, so we need to add back in these cases. Of course,
students both not receiving an award is equivalent to only
student receiving all
awards, so there are simply
choices for which student that would be. Therefore, the total number of ways of distributing the awards is
.
Solution 2 (Constructive Counting)
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, two student receives awards and one student recieves
award . We know there are
choices for which student gets
award. There are
ways to do this. Then, there are
ways to give the first student his two awards, leaving
awards yet to distribute. There are then
ways to give the second student his
awards. Finally, there is only
student and
award left, so there is only
way to distribute this award. This results in
ways to distribute the awards in this case. Adding the results of these two cases, we get
.
Video Solution
Subscribe!
Video Solution by WhyMath
~WhyMath
Video Solutions by The Learning Royal
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=1443
~Interstigation
Video Solution by STEMbreezy
https://youtu.be/wq8EUCe5oQU?t=243
~STEMbreezy
See also
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.