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[red_ring_of_box]

that's because the actual dashboard and application etc probably take up the 5 gb your not seeing I'd try either upgrading to a larger drive for more space or just dealing with it  

Heres a site where a list is kept of all the drives are available that are known to be lockable with xbox softmods:

http://xboxdrives.x-pec.com/?p=list

If you have a softmodded xbox as I do you will NEED to have a lockable drive to get it to work in the xbox.(I'm only saying this because I have no idea if you installed the drive or bought the xbox like this.)

 for a modchip install I don't know. I'm, sure Heimdall or someone else with experience with modchips can give you a better idea of what to do with a modchip install, though I know that the drives do not need to be locked with a working modchip installed.

[Heimdall]

Looks perfectly normal to me. The stock partitions take 8G:

C: 500MB
E: 4895MB
X: 750MB
Y: 750MB
Z: 750MB

That totals 7645 MB. The 11441MB on your F drive brings it up to 19086 MB. The difference between that and the 20GB you were expecting is because 1MB does not equal 1 million bytes - see this link for details of why manufacturers quoted disk sizes are larger than those reported by the operating system.

The stock 5960 MS dashboard takes roughly 170 MB on the C drive, leaving around 330MB free. If you have a recent softmod you aren't seeing the raw C, you're looking at the shadow C, which is roughly 330 MB when empty.

[Bomb Bloke]

Note that the X/Y/Z partitions are used for caching, and the console will automatically wipe their contents on occasion.

And before you ask, no, you can't merge/resize some of these partitions to get more space.