The question asks "How many times will the digit 7 be written when listing the integers from 1 to 1000?" not how many number will have 7. So, for example, for number 777, it should be counted as three 7's not one.
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Since neither 0 not 1000 has 7 in it, then excluding/including these values into the range won't change the answer.
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Hi..
if we write all as 000 to 999, we will have 3000 digits as 3(number of digits in each number )*1000(total numbers)=3000..
In these 3000, all 10 digits will be 3000/10=300..

But we do not have 0s in single digit and two digits..
so 000,as we do not have 0, so 3 0s missing
00a.. from 1 to 9, they are not written as 001,002...009, so 2*9=18 x 0s missing
0ab.....from 10 to 99, they are not written as 010,011...099, so 1*90=90 x 0s missing
Total 3+18+90=111 number of 0s are less..

Thus, from 1 to 999, we have 300 of all 9 digits - 1 to 9, while we have 300-111=189 of 0s.
Add 1000,
digit 0 is used 189+3=192 times
digit 1 is used 300+1=301 times
digits 2 to 9, 300 times each
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