In the expansion of $(a+b+c)^{20}$, find the coefficient of $a^{17} b^2c$

Question

In the expansion of $(a+b+c)^{20}$, find the coefficient of $a^{17} b^2c$

$C\binom {20} {3}$ $(b+c)^3$

which gives $a^{17}[b^3+c^3+bc(b+c)]$

$a^{17}[b^3+c^3+b^2+bc^2]$

Coefficient of $a^{17}b^2c$ is $C\binom {20} {3}$ which is equal to $1140$

But the answer is $3420$.

How?

Where am I doing mistake?

Answer 1

$$(b+c)^3=b^3+\color{red}{3}b^2c+\color{red}{3}bc^2+c^3$$

Hence the answer is $\binom{20}{3} \binom{3}{1}$.