In how many of the integer numbers between 0 and 10,000 does the digit 3 appear to the left of 4 In how many of the integer numbers between $0$ and $10\,000$ does the digit $3$ appear some place to the left of the digit $4$? This would include, for example, the numbers $34$, $374$, $4384$ and $3874$, but would not include $27$, $43$, $3650$ or $4333$.
So I am thinking you do $$8\cdot9\cdot1\cdot1+1\cdot9\cdot1+8\cdot1\cdot9\cdot1+1\cdot9\cdot9\cdot1 = 234$$
But I feel like that is too low.

Revised answer
Marking the first $3$ and the first $4$ after $3$, there are only $6$ cases:
$$\begin{array}{}
3&4&\bullet&\bullet&\implies& 10\cdot10 = 100\\
3&\bullet&4&\bullet&\implies& 9\cdot 10 = 90\\
3&\bullet&\bullet&4&\implies& 9\cdot9 =81\\
\bullet&3&4&\bullet&\implies& 9\cdot10 = 90\\
\bullet&3&\bullet&4&\implies& 9\cdot 9 = 81\\
\bullet&\bullet&3&4&\implies& 9\cdot9 = 81
\end{array}$$
Adding up, $523$.
Note: The first $4$ after the first $3$ has been marked, but this doesn't preclude a $4$ coming before the first $3$.
