TITLE: Choosing books on a bookshelf where no two are consecutive
QUESTION [9 upvotes]: I'm going through Cohen's Basic Techniques of Combinatorial Theory and I am stuck on this problem:
There are 24 volumes of an Encyclopedia on a bookshelf. In how many ways can 5 of these books be selected if no 2 consecutive volumes are to be chosen?
Any help would be much appreciated! 
Thanks

REPLY [3 votes]: I love to think about this problem this way:  
We start choosing by the lowest number, and will not choose any number before that number and I proceed by choosing only in increasing order.  
So, when we choose the lowest number, we eliminate the number next to the chosen number. We continue this process and on the final selection of number, We have chosen all $5$ numbers so we do not have to eliminate any number anymore.  
If one thinks carefully, then one will observe that we have eliminated $4$ numbers and we can choose any $5$ numbers from the remaining $20$ numbers as $20\choose 5$