  A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic
factors. If the bound $n+1$ is attained, the word $w$ is called rich. An
infinite word $w$ is called rich if every finite factor of $w$ is rich.
  Let $w$ be a word (finite or infinite) over an alphabet with $q>1$ letters,
let $F(w,n)$ be the set of factors of length $n$ of the word $w$, and let
$F_p(w,n)\subseteq F(w,n)$ be the set of palindromic factors of length $n$ of
the word $w$.
  We present several upper bounds for $| F(w,n)|$ and $| F_p(w,n)|$, where $w$
is a rich word. In particular we show that \[| F(w,n)| \leq
(q+1)8n^2(8q^{10}n)^{\log_2{2n}}+q\mbox{.}\]
  In 2007, Bal{\'a}{\v z}i, Mas{\'a}kov{\'a}, and Pelantov{\'a} showed that \[|
F_p(w,n)| +| F_p(w,n+1)| \leq | F(w,n+1)|-| F(w,n)|+2\mbox{,}\] where $w$ is an
infinite word whose set of factors is closed under reversal. We generalize this
inequality for finite words.
