Number of all 4 – digited integers formed with exactly two distinct digits.

Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
Select one digit o then other digit any one from 1 to 9 in 9 ways  
With these two digits number of 4 digited numbers 
 

One zero and other three digits non zero

Number of 4-digits numbers when 0 in is $9 + 27 + 27 = 63$
Now select any digits in $9_{c_2}$ ways  $\to \frac{9\times 8}{2}=36$
Say  $\to a,b$ 
$$\begin{gathered} aaab\to \frac{\overline{)4}}{\overline{)3}}=4 \\ abbb\to \frac{\overline{)4}}{\overline{)3}}=4 \end{gathered}$$

Number of ways a, b any of -14 way  $\Rightarrow$ Number ways 2-non two digits form 4-digited numbers is total no of ways $= 504 + 63= 567$