TITLE: Number of ways to choose 6 books out of 20 books such that no 2 are adjacent books
QUESTION [5 upvotes]: I was trying to do the following question:

Describe a bijection between ways of choosing 6 books out of 20 books
  so that no two adjacent books are selected and a 15-bit sequence with
  exactly 6 ones.

I attached a useful picture from the solutions:

The solutions actually nearly make sense to me, except one detail. It seems to me that it has a small mistake (or a maybe I have a misunderstanding) because not all the ones can actually be adjacent to each other. For example, if it happens that the last 1 and the first 1 are actually together then it could mean that we are selection first a book and then a pair (book, empty), but what would correspond to selecting two books that are adjacent.
For me intuitively it would make more sense if we would have described the sequence they suggested by:
100110001001002
where the digit 2 corresponds to the special case indicating that its not a pair, (book, no book) but in fact it corresponds to the singleton selection (book).
However, with that set up it makes it hard for me to count things cuz the 2 cannot be next 1's.... am i wrong or do the solutions have a small typo?

Also, why is it not undercounting? This is why I believe it is undercounting:
It seems to be undercounting (I think), cuz I could have choose a single book at the beginning and the reverse the pairs thereafter. i.e. (book),(no book, book), ..., (no book, book). How is the given solution not missing that sequence?

REPLY [2 votes]: You want a bijection between the $15$ bit sequences with excactly $6$ ones and the $20$ bit sequences that don't contain the sub-sequence $11$. The bijection is to add a zero to the right of every $1$ except for the right-most one.

Another approach via stars and bars, you want to distribute $14$ zeroes between the $6$ ones so that the $5$ spaces in the middle have at least one zero each, so in reality there are only $9$ zeroes to distribute with the $6$ ones as separators. The answer is hence $\binom{15}{6}$