Combination Problem Understanding How many ways  can a Doctor go to the Hospital on  $5$ days of January (which has $31$ days) such that no two visits are on consecutive days?
I think the solution is: $\displaystyle\binom{27}{5}$
But I have  no idea if I'm right. 

The argument goes: after the first four hospital days ($H$), there must be a non-hospital day ($NH$).  So we find a permutation of $$\{H,H,H,H,H,\overbrace{NH,NH,\ldots,NH}^{27}\}$$ of which there are $\binom{27}{5}$ then add $NH$ after the first four $H$s.

For example, the permutation might come out as:
$$(NH,NH,NH,NH,NH,NH,NH,\color{blue}{H},\color{red}{H},NH,NH,NH,NH,NH,NH,NH,NH,NH,NH,\color{green}{H},NH,NH,\color{orange}{H},NH,NH,NH,NH,NH,NH,NH,H,NH)_{27 \text{ days}}$$
which has a clash initially, but no longer after we insert $NH$s after the first four $H$s:
$$(NH,NH,NH,NH,NH,NH,NH,\color{blue}{H},\color{blue}{NH},\color{red}{H},\color{red}{NH},NH,NH,NH,NH,NH,NH,NH,NH,NH,NH,\color{green}{H},\color{green}{NH},NH,NH,\color{orange}{H},\color{orange}{NH},NH,NH,NH,NH,NH,NH,NH,H,NH)_{31 \text{ days}}.$$
