Answer
$q^9 + 9 q^8 r + 36 q^7 r^2 + 84 q^6 r^3 + 126 q^5 r^4 + 126 q^4 r^5 + 84 q^3 r^6 +\\ 36 q^2 r^7 + 9 q r^8 + r^9
$
Work Step by Step
Using the Binomial Formula, the expression $
(q+r)^9
$ expands to
\begin{array}{l}
q^9r^0+
\dfrac{9}{1!}q^8r^1+
\dfrac{9\cdot8}{2!}q^7r^2+
\dfrac{9\cdot8\cdot7}{3!}q^6r^3+
\dfrac{9\cdot8\cdot7\cdot6}{4!}q^5r^4+\\
\dfrac{9\cdot8\cdot7\cdot6\cdot5}{5!}q^4r^5+
\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot4}{6!}q^3r^6+
\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3}{7!}q^2r^7+\\
\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2}{8!}q^1r^8+
\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1}{9!}q^0r^9
\\\\\\=
q^9 + 9 q^8 r + 36 q^7 r^2 + 84 q^6 r^3 + 126 q^5 r^4 + 126 q^4 r^5 + 84 q^3 r^6 +\\ 36 q^2 r^7 + 9 q r^8 + r^9
\end{array}
