(let a partition of $n \in \Bbb N, n>0 $ be a sum of positive integers)

How would you show that the number of different partitions of $3$ is the coefficient of 
$x^3$
in the
expansion of:

$(1+x+x^2+x^3)(1+x^2)(1+x^3)?$

My attempt:

since there are $3$ brackets to choose factors of $x^3$ from, we can say the following:

$ x^3 = x^a x^b x^c, a \in \{0,1,2,3\}, b \in \{0,2\}, c \in \{0,3\}$

The coefficient of $x^3$ in the expansion will equal the number of solutions satisfying $ x^3 = x^a x^b x^c$.

$a=0 \Rightarrow b=0 $ and $ c=3$,

$a=1 \Rightarrow b=2 $ and $ c=0$,

$a=2 \Rightarrow$ no solution.

$a=3 \Rightarrow b=0 $ and $ c=0$,

$\Rightarrow$ the coefficient of $x^3$ is $3$ which equals the number of partitions of $3$.

Do I understand what the question is asking? Because I'm wondering if just multiplying the brackets and obtaining the coefficient is an equivalent solution to the question.

Thanks.