Problem 25 · 2020 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
At the Sunday fair, in the morning Ana wanted to buy three kinds of fruit out of 12 options and one kind of vegetable out of the 6 available. In the afternoon some products had sold out, and Bela wanted to buy two kinds of fruit and two kinds of vegetable from those remaining. Since the number of possible choices for Bela was a quarter of the number of possible choices for Ana, how many products had sold out by the afternoon?
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Answer: C — 3
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Hint 1 of 2
Write Ana's count C(12,3)·C(6,1) and Bela's count with reduced stocks.
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Hint 2 of 2
Set Bela's count to a quarter of Ana's and solve.
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Approach: set the two combination counts in a 1:4 ratio
- Ana has C(12,3)·C(6,1) = 1320 choices.
- After f fruits and v vegetables sell out, Bela has C(12−f,2)·C(6−v,2), which must equal 1320/4 = 330.
- The only solution is f = 1, v = 2, so the number sold out is 3.
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