Numbers with at least one digit 7

Text

How many integers from $1$ to $9999$ contain at least one digit $7$?

Solution

We count by complement, representing each integer from $0$ to $9999$ as a $4$-digit string (with possible leading zeros, from $0000$ to $9999$): these are $10^{4}=10\,000$ strings. The strings that do not contain any $7$ use only $9$ digits in each position: $9^{4}=6561$. So the strings with at least one $7$ number $10^{4}-9^{4}=3439$. Since the excluded string $0000$ (which corresponds to no integer between $1$ and $9999$) contains no $7$, it does not affect this count: $N = 10^{4} - 9^{4} = 10\,000 - 6561 = \boxed{3439}.$