Problem 320

Since 100 000 does not include the block of digits 178, each
integer between 10 000 and 100 000 that includes the block of digits 178
has five digits.

Such an integer can be of the form $17\,8xy$ or of the form $x1\,78y$ or of the form $xy\,178$ for some digits $x$ and $y$.

The leading digit of a five-digit integer has 9 possible values (any
digit from 1 to 9, inclusive) while a later digit in a five-digit
integer has 10 possible values (0 or any digit from 1 to 9,
inclusive).

This means that

there are 100 integers of the form $17\,8xy$ (10 choices for each of $x$ and $y$, and $10
$\times$ 10 = 100$),
there are 90 integers of the form $x1\,78y$ (9 choices for $x$ and 10 choices for $y$, and $9
$\times$ 10 = 90$), and
there are 90 integers of the form $xy\,178$ (9 choices for $x$ and 10 choices for $y$, and $9
$\times$ 10 = 90$).

In total, there are thus $100 + 90 + 90 =
280$ integers between 10 000 and 100 000 that include the block
of digits 178.