Vord-search

Let $V(n)$ be the number of vords with a total value of $n$ .

The number of distinct permutations of $p$ "A"s, $q$ "B"s and $r$ "C"s is $\frac{(p+q+r)!}{p!q!r!}$ .

So we have

$V(n)=\sum_{p+3q+4r=n} \frac{(p+q+r)!}{p!q!r!}$

(where $p,q,r$ are all non-negative integers). For computation, we can simplify this slightly by making sure $3q+4r \le n$ and replacing $p$ with $n-3q-4r$ .

$V(n)=\sum_{3q+4r \le n} \frac{(n-2q-3r)!}{(n-3q-4r)!q!r!}$

Finally, the number of vords with value $14$ that end in the letters "BC" is just $V(7)$ . So for the answer we compute $V(14)-V(7)=441-15=\boxed{426}$ .