Problem 15 · 2017 Math Kangaroo
Hard
Counting & Probability
complementary-counting
The four faces of a regular tetrahedron are labelled with the four digits 2, 0, 1 and 7 (one digit on each face). For a game, four such tetrahedrons are used as fair dice. All four dice are thrown simultaneously. Three of the four faces of each die can then be seen from above. What is the probability that we can form the number 2017 using exactly one of the three visible digits of each die?
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Answer: B — 6364
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Hint 1 of 2
On each die the three visible faces are simply all faces except the one hidden underneath.
Still stuck? Show hint 2 →
Hint 2 of 2
It is easier to count the chance you CANNOT form 2017, then subtract from 1.
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Approach: complementary counting over which digit each die hides
- Each die hides exactly one of its four digits; the other three are visible. There are 4^4 = 256 equally likely hidden-digit combinations.
- You can match the four needed digits (2,0,1,7) to the four dice unless some required digit is hidden on every die.
- That fails only when all four dice hide the same one digit: 4 ways out of 256.
- So the probability of success is 1 - 4/256 = 63/64.
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