## 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: $3050$