  A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert
u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the
border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is
privileged if $\vert u\vert\leq 1$ or if $u$ contains a privileged border $w$
that appears exactly twice in $u$.
  Peltom\"aki (2016) presented the following open problem: "Give a nontrivial
upper bound for $B(n)$", where $B(n)$ denotes the number of privileged words of
length $n$. Let $D(n)$ denote the number of closed words of length $n$. Let
$q>1$ be the size of the alphabet. We show that there is a positive real
constant $c$ such that \[D(n)\leq c\ln{n}\frac{q^{n}}{\sqrt{n}}\mbox{, where
}n>1\mbox{.}\]
  Privileged words are a subset of closed words, hence we show also an upper
bound for the number of privileged words.
