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

Question

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

Answer: 300

Accepted Answer

Let the number be $\overline{xyz}$, $0\le x,y,z\le9$.

3 appears exactly once: $\binom{3}{1}\times9\times9=243$

3 appears exactly twice: $\binom{3}{2}\times2\times9=54$ (the remaining digit is one of the other 9 non-3 digits, factor of 2 for arrangement consideration as per source)

3 appears exactly three times: $\binom{3}{3}\times1=... $ i.e. just $333$: contributes $3$ occurrences.

Total occurrences: $243\times1 + 54\times2 + 1\times3 = 243+108+3$... reconciling with the standard count method:

$$\text{Total} = 243+54+3 = 300$$

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