There are : (200 - 1) : 1 + 1 = 200 numbers from 1 to 200

We have a sequence of two-digit numbers are as follows :

2 ; 12 ; 22 ; 32 ; ....... ; 192

Apply a formula to count numbers , we have :

(192 - 2) : 10 + 1 = 20 

There are also numbers with 2 digits such as

20 ; 21 ; 23 ; 24 ; .... ; 29 ; 120 ; 121 ; 122 ; 123 ; ...... 129

And there are have 18 also numbers .

So , from these information , we have 38 numbers have digit 2

We have :

200 - 38 = 162 (numbers)

There are 162 numbers from 1 to 200 do not have the digit 2 .