Here are some statistics as to the diameter of the smallest circle that $n$ diameter-$1$ circles can be packed into:
$$\begin{matrix}
2 & 2 \\
3 & 2.1547 \\
12 & 4.0296 \\
13 & 4.2361 \\
31 & 6.2915 \\
32 & 6.4295 \\
50 & 7.9475\end{matrix}$$
The best-known packing of $31$ puts $1$ in the inner $2.1$-circle, $6$ others in the $4.2$-circle and the other $31-7=24$ in the $6.3$-circle. The other $50-31=19$ fit in the outer ring. This gives a score of $1\cdot30+6\cdot20+24\cdot14+19\cdot5=581$.
If, alternatively, we pack as many as possible into the $4.2$-circle, we pack $12$ there. Of those $12, 3$ are as near to the centre as possible, but fail to fit into the inner $2.1$-circle, so each of the $12$ scores $20$. $16$ more will fit in the $6.3$-circle, though $17$ won't, because doing that entails packing their centres onto a $5.3$-circle, and $5.3\pi\approx16.65<17$. $50-12-16=22$ fit in the outer ring. This gives a score of $12\cdot20+16\cdot14+22\cdot5=574$ so this is worse.
The ratios between the scores are carefully chosen. If circles in the $6.3$-circle scored only $13$ instead of $14$, $581$ and $574$ would be $557$ and $558$, so the formerly worse solution would then be better.
The results for the smallest circle into which $n$ unit circles can be packed come from Graham[1].
I could've sworn that all the dimensions specified were diameters.
12 radius-$1$ circles pack into a circle of radius $4.0296$. Therefore 12 diameter-$1$ circles pack into a circle of radius $2.1$. Now consider the second ring (of radius $r=4.2$). $2\pi (r-\frac12)>23$ and $2\pi (r-1\frac12)>16$, so there is room in the second ring for a ring of $15$ or $16$ and a ring further out of $22$ or $23$. The score would then be $12\cdot30+38\cdot20=1120$. But there is no subtlety here, and the largest two of the four concentric rings are not needed. I think the intended problem must be as I originally thought it was: with circles of diameter $1$ in rings of diameter $2.1, 4.2, 6.3, 8.4$. Or circles of radius $1$ in rings of radius $2.1, 4.2, 6.3, 8.4$.
[1] Graham et al., Dense packings of congruent circles in a circle. Discrete Mathematics 181 (1998), p.139-154.
