The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

 Any number between 1 to 999 is a 3,digit number xyz 

where the digits xyz are any digits from 0 to 9.
are any digits from 0 to 9.
Now, we first count the numbers in which 3 occurs once
only- Since, 3 can occur at one place in ${ }^3{\mathrm{C}}_1$, ways, there
are${ }^3{\mathrm{C}}_1\cdot (9\times 9)=3\cdot 9^2$ such numbers

$3_{C_1}{9}^2+2\times 3_{C_2}\times 9+3\times 3_{c_3}=243+54+3=300$