I'm working through some combinatorics practice sets and found the following problem that I can't make heads or tails of.

It asks to prove the following:

$$128! = \binom{128}{64}\binom{64}{32}^2\binom{32}{16}^4\binom{16}8^8\binom 84^{16}\binom 42^{32}\binom{2}{1}^{64}$$

Weird, huh? The first thing I noticed is that the exponents mirror the $r$ variables. I would normally just re-express each statement in $\frac{n!}{(n-r)!r!}$ form, but the exponents throw me for a loop. Are there any intuitions about factorials or nCr I should be considering here?