Tessellate S.T.E.M.S - Mathematics - College - Set 2 - Problem 5

Let's consider $Comb(A)$ the set of all the ${16}\choose{5}$ possible combinations with no repetition of $A=\{1,...,16\}$ . The $good$ sets are $(a_1,...,a_5)\in Comb(A)$ such that $a_1 \in [1,2,3,4]$ , $a_5\in [13,14,15,16]$ and $1< a_k-a_{k-1} \leq 4$ , $k \in \{2,3,4\}$ . With such condition, it's possible to code an algorithm to count all possible occurrences.

The answer is the coefficient of $x^{11}$ in the product $(1+x+x^2+x^3)(x+x^2+x^3)^4(1+x+x^2+x^3).$ The first polynomial and the last polynomial represent the choices of a number of books before the first book in the set, and a number of books after the last book in the set, respectively; the middle polynomial represents the four choices for the numbers of books in between each pair of books in the set (there is no constant term because the set cannot contain consecutive books).

Simplifying a bit, this is the coefficient of $x^7$ in the product $(1+x+x^2+x^3)^2(1+x+x^2)^3.$ To be honest, I just multiplied this out using a computer algebra package. Perhaps there is a fancy way to do it. Anyway, I got $220.$