College Algebra, Chapter 9, 9.6, Section 9.6, Problem 40

Determine the term containing $y^3$ in the expansion of $(\sqrt{2} + y)^{12}$

The term that contains $a^r$ in the expansion of $(a + b)^n$ is

$\displaystyle \left( \begin{array}{c}
n \\
n - r
\end{array} \right) a^r b^{n - r}$

If we rewrite the expansion as $(y + \sqrt{2})^{12}$, we have $a = y, b = \sqrt{2}, n = 12$ and $r = 3$, so the term containing $y^3$ is


$
\begin{equation}
\begin{aligned}

=& \left( \begin{array}{c}
12 \\
12 - 3
\end{array} \right) y^3 (\sqrt{2})^{12 - 3}
\\
\\
=& \left( \begin{array}{c}
12 \\
9
\end{array} \right) y^3 (\sqrt{2})^9
\\
\\
=& \frac{12!}{9! (12 - 9)!} (2)^{\frac{9}{2}} y^3
\\
\\
=& 220 (2)^{\frac{9}{2}} y^3
\\
\\
=& 220 (2)^{\frac{1}{2}} (2)^4 y^3
\\
\\
=& 3520 \sqrt{2} y^3



\end{aligned}
\end{equation}
$