Problem 20 · 2011 Math Kangaroo
Hard
Counting & Probability
careful-countingcasework
You can place together the cards pictured to make different three‑digit numbers, for instance 989 or 986. How many different three‑digit numbers can you make with these cards?
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Answer: E — 12
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Hint 1 of 3
Two of the cards are the kind that show a 6 one way and a 9 when you turn them upside down; the 8 looks the same either way.
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Hint 2 of 3
First decide which of the three spots the 8 sits in, then fill the other two spots.
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Hint 3 of 3
For each place the 8 can go, the two leftover spots can each be a 6 or a 9.
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Approach: place the 8, then fill the other two spots with 6 or 9
- There are three cards: two flip-cards that each show a 6 or a 9, and one 8-card that looks the same whichever way up it is.
- Pick where the 8 goes: it can be the first, middle, or last digit — that is 3 choices.
- For each of those, the two remaining spots can each be a 6 or a 9, giving 4 numbers per choice: for example with 8 in front you get 866, 869, 896, 899.
- So the count is 3 groups of 4, which is 3 × 4 = 12 different numbers, answer E.
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