Burt Kaliski Jr. (@modulomathy)

3/6/12 7:14 AM

#math trivia for #March6: #66 is “two of a kind”. What are the odds of getting “two of a kind” in a leap year? (Ignore leading 0s)

The question is asking for the number of integers between 1 and 366 that have exactly two of some digit in their decimal representation. The smallest such integer is 11; it is easy to see that there are nine such integers between 1 and 99.

Between 100 and 199, how many are there? The integer must have one of these forms:

- 1×1 where x is not 1 — nine choices
- 11x where x is not 1 — another nine choices
- 1xx where x is not 1 — another nine choices

There are 27 “two of a kinds” in this range.

The number of “two of a kinds” between 200 and 299 is also 27 by the same analysis.

Between 300 and 366, there are — six of the form 3×3, nine of the form 33x, and six of the form 3xx, where x is not 3. This adds up to another 21 choices, or 84 overall. The odds of “two of a kind” are thus 84 out of 366, or 14 out of 61, which is just under one in four.

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