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. 

You can represent the 26 days when the doctor does not go to the hospital by 26 sticks, which create 27 gaps.  If you select 5 of these gaps, this determines the 5 days when the doctor does go in; so this shows that your answer of $\binom{27}{5}$ is correct.

Another way to see this is to let $x_1, \cdots, x_6$ be the number of days in the gaps created by the 5 visits.  
Then $x_1+\cdots+x_6=26$ where $x_1\ge0$, $x_6\ge0$, and $x_i\ge1$ for $2\le i\le5$.
If we let $y_i=x_i$ for $i=1,6$ and $y_i=x_i-1$ for $2\le i\le5$, 
then $y_1+\cdots+y_6=22$ with $y_i\ge0$ for each $i$, and
the number of integral solutions to this equation is given by $\binom{27}{5}$.
