The other day I came across a paper that addresses a closely related question:

http://www.unige.ch/math/folks/velenik/Vulg/Paninimania.pdf

If I have understood it correctly, the expected number of packs you would need to buy would be:

$\binom{424}{5}\sum_{j=1}^{424}\left(-1\right)^{j+1}\frac{\binom{424}{j}}{\binom{424}{5}-\binom{424-j}{5}}$

However, as eqperes points out in the comments, the specific question the OP asks is actually covered in detail in another paper that is not open access.