2019 AMC 8 Problems/Problem 25
Contents
[hide]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. The problem asks for the number of integer solutions such that
and
. We can subtract 2 from
,
,
, so that we equivalently seek the number of non-negative integer solutions to
. By stars and bars (using 18 stars and 2 bars), the number of solutions is
.
Solution 2
Let's say you assume that Alice has 2 apples. There are 19 ways to split the rest of the apples with Becky and Chris. If Alice has 3 apples, there are 18 ways to split the rest of the apples with Becky and Chris. If Alice has 4 apples, there are 17 ways to split the rest. So the total number of ways to split 24 apples between the three friends is equal to 19 + 18 + 17...…… + 1 = 20 (19/2) =
~heeeeeeheeeeeee
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
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