I think the provided answer to the problem 23 of Chapter 1 is not considering the given condition that any button won't be pressed more than once.
... Each presses the button for their desired floor (unless one of the others has already pressed that button)...
So, I think the answer should be this:
$$\frac { 7 }{ \binom{ 9 }{ 1 } + \binom{ 9 }{ 2 } + \binom{ 9 }{ 3 } } = \frac{ 7 }{ 129 }$$
I think the provided answer to the problem 23 of Chapter 1 is not considering the given condition that any button won't be pressed more than once.
So, I think the answer should be this: