Problem 18 · 2001 AMC 8
Hard
Counting & Probability
complementary-counting
Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5?
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Answer: D — 11/36.
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Hint 1 of 2
A product carries a factor of 5 only if one of the dice supplies it — and the only multiple of 5 on a die is 5 itself. So the event is really "at least one die shows a 5."
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Hint 2 of 2
"At least one" is the cue to flip to the complement: count the chance that NEITHER die is a 5, which is one clean multiplication.
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Approach: complementary counting
- Where can the factor of 5 come from? Only the face 5 (since 10, 15, … aren't on a die). So a multiple-of-5 product means at least one die is a 5.
- Counting "at least one" directly is messy, so count the opposite: P(neither is 5) = (5/6)(5/6) = 25/36. Then P(at least one 5) = 1 − 25/36 = 11/36.
- Whenever you see "at least one," reach for the complement — "none" is usually a single product instead of a pile of overlapping cases.
Another way — count the favorable cells directly:
- Of 36 equally likely (die1, die2) outcomes, a 5 appears in row 5 (6 cells) or column 5 (6 cells).
- That's 6 + 6 − 1 = 11 cells (the (5,5) overlap counted once), giving 11/36.
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