There are two methods.  I'll do both.

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The largest 10 digit number that has the first three digits 407 is

4079999999

and that number is the the number of positive integers from 1 to 4079999999,
inclusively.

We wish to subtract from that the number of integers less than that number
which do not have 407 as its first three digits.

The largest 10-digit number that does not have its first three digits as 407 is

4069999999

and that is the number of positive integers from 1 to 4069999999, inclusively.

So we subtract the second number from the first number:

 4079999999
-4069999999
-----------
   10000000

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Or we could do it as an arithmetic sequence:

, , 

Then use the formula











Edwin