Problem 7 · 2020 Math Kangaroo
Medium
Counting & Probability
careful-countingcasework
The digits of the year 2020 are each repeated twice and the two digits are different. How many times does this happen between the year 1000 and now (2020)?
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Answer: E — 29
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Hint 1 of 2
You want years like 2020 made of two different digits, each appearing exactly twice.
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Hint 2 of 2
Split by the leading digit: count such years starting with 1, then those starting with 2 up to 2020.
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Approach: count the two-distinct-digit, each-twice years by leading digit
- Such a year uses two different digits, each appearing twice, e.g. 1331 or 2020.
- Years from 1000–1999 (leading 1): the other digit pairs with one more 1 plus two of a different digit; enumerating the patterns gives 27 of them.
- Years from 2000–2020 add just 2002 and 2020, for 2 more.
- Total = 27 + 2 = 29, choice E.
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