How many three-digit positive integers have at least one prime digit?

Here's my best attempt  (I'm not great at these counting problems !!! )

Non-prime digits = (0, 1, 4, 6, 8, 9 )

Number of three digit numbers not having any prime digits

Hundreds place  = (Choose  any  one of the digits 1,4,6,8,9) =  C(5,1) = 5

Tens place  = ( Choose any one of the digits 0,1,4,6,8,9) = C(6,1)  = 6

Ones place = (Choose any one  of the digits 0,1,4,6,8,9) = C(6,1)  = 6

So   5 * 6 * 6 =    180 have no prime digits

So

900  three digit numbers -  180 with no prime digits  =    

720  have at least one prime digit

There are \(900 - 5 \times 6 \times 6 = \color{brown}\boxed{720}\) numbers with at least 1 prime digit.