One way is to consider the digit combinations that allow for "two the same" while also being "positive three-digit primes less than $200$":
After finding all possible digit combinations (are there any missing from the above list?), it should be relatively easy to find out which numbers matching one of these categories are prime.
To conclude this approach, consider using $X\in\{0,1,2,3,4,5,6,7,8,9\}$, where we get
$\{101,111,121,131,141,151,161,171,181,191\}\to\{101,131,151,181,191\}$
Using $Y,Z\in\{1,3,7,9\}$, we get
$\{111,113,117,119,133,177,199\}\to\{113,199\}$
Except for $121$, the smallest prime factor of any of the $18$ digit-combination candidates was not greater than $7$, and the final list of primes meeting the stated criteria is:
$\{101,113,131,151,181,191,199\}$