Ambiguous Tickets

100,\!000 raffle tickets have been printed, each with a distinct 5-digit number ranging from 00000 to 99999. However, some tickets are ambiguous, depending on the ticket’s orientation. For example, is the ticket below 0908\text{I} or \text{I}8060? How many of the 100,\!000 tickets are ambiguous?

Source: NCTM Mathematics Teacher 2008

SOLUTION
Examples of unambiguous numbers
00000,0\text{I}0\text{I}0,06090,\cdots
00\text{I}00,0\text{III}0,06\text{I}90,\cdots
00800,0\text{I}8\text{I}0,06890,\cdots
Characteristics of an unambiguous number
1) the third digit remains the same after the rotation: 0,\text{I},8
2) once the first two digits are chosen, there is only one way to choose the last two digits.
How many unambiguous numbers?
5\times 5\times 3\times 1\times 1=75
How many numbers?
5\times 5\times 5\times 5\times 5=3125
How many ambiguous tickets?
3125-75=3050

Answer: 3050

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About mvtrinh

Retired high school math teacher.
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