How many of the 9000 four digit integers have four digits that are increasing? How to find the number of distinct four digit numbers that are increasing or decreasing? 
The correct answer is $2{9 \choose 4} + {9 \choose 3} = 343$. How to get there?

The analysis that was used goes as follows:
Not using $0$: We choose $4$ non-zero digits. Once we have done that, we can arrange them in increasing order in $1$ way, and in decreasing order in $1$ way, for a total of $2\binom{9}{4}$.
Using $0$: They can only be decreasing. And we need to choose $3$ non-zero digits to go with the $0$. This can be done in $\binom{9}{3}$ ways.
I prefer Juanito's approach. Note that the sum is not $343$, so if the book got that, there is a computational error.
