You have a string of $4$ digits. work out the number of strings in which $0$ and $1$ cannot be included at the same time.
In the solution it gave $5040-[(4)(3)(8)(7)]$.
I understand where $5040$ comes from.
However, I do not understand where $(4)(3)$ comes from. I understand this is from working out all the instances of $0$ and $1$ in the string, but how do we derive the $(4)(3)$?
You didn’t say so, but in order for the $5040$ to make sense, we must be talking about strings of distinct digits, so that we’re starting with $10\cdot9\cdot8\cdot7=5040$ possible strings. Now, as you say, we want to count the strings that contain both $0$ and $1$. There are $4$ positions in which we can put the $0$, and after we’ve done that, there are $3$ positions in which we can put the $1$. Then there are $8$ possible digits for the first of the two remaining places and $7$ possible digits for the last unfilled place, so there are altogether $4\cdot3\cdot8\cdot7$ of these strings that contain $0$ and $1$.