Choice Problem: choose 5 days in a month, consecutive days are forbidden I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. 

Example 3.19. A medical student has to work in a hospital for five
  days in January. However, he is not allowed to work two consecutive
  days in the hospital. In how many different ways can he choose the
  five days he will work in the hospital?
Solution. The difficulty here is to make sure that we do not choose two consecutive days. This can be assured by the following
  trick. Let $a_1, a_2, a_3, a_4, a_5$ be the dates of the five days of
  January that the student will spend in the hospital, in increasing
  order. Note that the requirement that there are no two consecutive
  numbers among the $a_i$, and $1 \le a_i \le 31$ for all $i$ is equivalent
  to the requirement that $1 \le a_i < a_2 — 1 < a_3 — 2 < a_4 — 3 < a_5 — 4 \le 27$. In other words, there is an obvious bijection between the
  set of 5-element subsets of [31] containing no two consecutive
  elements and the set of 5-element subsets of [27].
*** Instead of choosing the numbers $a_i$, we can choose the numbers $1 \le a_i < a_2 — 1 < a_3 — 2 < a_4 — 3 < a_5 — 4 \le 27$, that is, we can
  simply choose a five-element subset of [27], and we know that there
  are $\binom{27}{5}$ ways to do that.

What I don't understand here $1 \le a_i < a_2 — 1 < a_3 — 2 < a_4 — 3 < a_5 — 4 \le 27$:


*

*Why do the subtracting numbers increment with every other $a_i$? 

*Why 27?


And the very last sentence (***) is unclear to me.


*

*Why is there no talk about "non-consecutive"? Why choosing 5 elements of 27 is equivalent to choosing 5 non-consecutive elements out of 31? I miss the connection. 


I'd be very thankful if you could help me to understand this example!

These kind of problems can be generalized to a single formula in the following manner:
Suppose I have $n$ objects arranged in a row and I pick any $r$ objects such that no two of them are consecutive. I want to calculate in how many ways I can do this. To do that I first randomly select $r$ objects and take them out. Now there are $n-r$ objects left in the line and as a consequence there are $n-r+1$ gaps that can be filled between these objects. The number of ways in which I can fill these gaps is $\binom{n-r+1}{r}$. Since selection and deselection are equivalent in combinatorics, the answer to my problem is this only.
Now coming to your problem, in January there are 31 days and you have to select any 5 non-consecutive ones. The formula says that the answer would be:$$\binom{31-5+1}{5}$$which is nothing but $\binom{27}{5}$.
I'm sorry for not adhering to the solution in your book but I really think that this way is easier to understand.
