my english is not too good so exuse me.
This guide is for Samsung Galaxy Tab 3 10.1 P5200/P5210 only.
i take not responsibility for bricking your device so be carefull.
i have rooted my galaxy tab 3 10.1 P5200 without pre-rooted firmware and i want to show you how i did:
1-Download TWRP "Thanks to Angel_666":
2-Download Odin 3.07:
3-Download the ROOT file "thanks to DutchDanny,koush & cyanogenmod team"
UPDATE New link:http://www59.zippyshare.com/v/96195148/file.html
4-Reboot into download mode(hold volume down+power,you will see a warning message,press volume up to continune)
5-Open Odin,Connect the Tab to the computer and click on PDA,then select the TWRP recovery,
-make sure to uncheck 'auto-reboot' before flashing,because the tab might reinstall the stock recovery(thank you Benzoman for remind me).
6-When done,copy the ROOT file to the tab and reboot into recovery (hold volume up+power).
7-select install zip from sdcard and choose the root file.
8-flash the root file
9-suprise your device is rooted.
Have a nice day!!