2019 AMC 8 Problems/Problem 25
Contents
Problem 25
Alice has apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
Solution 1
We use stars and bars. Let Alice get apples, let Becky get
apples, let Chris get
apples.
We can manipulate this into an equation which can be solved using stars and bars.
All of them get at least apples, so we can subtract
from
,
from
, and
from
.
Let
, let
, let
.
We can allow either of them to equal to
; hence, this can be solved by stars and bars.
By Stars and Bars, our answer is just .
Solution 2
First assume that Alice has apples. There are
ways to split the rest of the apples with Becky and Chris. If Alice has
apples, there are
ways to split the rest of the apples with Becky and Chris. If Alice has
apples, there are
ways to split the rest. So the total number of ways to split
apples between the three friends is equal to
Solution 3
Let's assume that the three of them have apples. Since each of them has to have at least
apples, we say that
and
. Thus,
, and so by stars and bars, the number of solutions for this is
- aops5234
Solution 4
Since we have to give each of the friends at least
apples, we need to spend a total of
apples to solve the restriction. Now we have
apples left to be divided among Alice, Becky, and Chris, without any constraints. We use the Ball-and-urn technique, or sometimes known as ([Sticks and Stones]/[Stars and Bars]), to divide the apples. We now have
stones and
sticks, which have a total of
ways to arrange.
~by sakshamsethi
Solution 5
Equivalently, we split apples among
friends with each having at least
apples. We put sticks between apples to split apples into three stacks. So there are 20 spaces to put
sticks. We have
different ways to arrange the two sticks. So, there are
ways to split the apples among them.
~by Dolphindesigner
Solution 6 (Answer Choices)
Consider an unordered triple where
and
are not necessarily distinct. Then, we will either have
,
, or
ways to assign
,
, and
to Alice, Becky, and Chris. Thus, our answer will be
for some nonnegative integers
. Notice that we only have
way to assign the numbers
to Alice, Becky, and Chris when
. As this only happens
way (
), our answer is
for some
. Finally, notice that this implies the answer is
mod
. The only answer choice that satisfies this is
. -BorealBear
Video Solutions
https://www.youtube.com/watch?v=OPFJ-d1byw4 Math is cool
https://www.youtube.com/watch?v=EJzSOPXULBc - Happytwin
https://www.youtube.com/watch?v=wJ7uvypbB28
https://www.youtube.com/watch?v=2dBUklyUaNI
https://youtu.be/ZsCRGK4VgBE ~DSA_Catachu
https://www.youtube.com/watch?v=3qp0wTq-LI0&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=7 ~ MathEx
https://youtu.be/5UojVH4Cqqs?t=5131 ~ pi_is_3.14
~savannahsolver
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
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