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 all how i did:
1-Download TWRP Recovery (make sure to use the version that matches your tab model):
Link for the P5200 model: https://www.androidfilehost.com/?fid=23991606952613332
Link for the P5210 model: https://www.androidfilehost.com/?fid=23991606952613346
2-Download Odin 3.07:
3-Download the ROOT file (SuperSU):
UPDATE New link and version:
Incase if the SuperSU root method doesn't work,try this one instead (Superuser):
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 telling me that).
6-When done,copy the ROOT file direcly to the internal storage of the tab (or even external sdcard) and reboot into recovery (hold volume up+power).
7-select install zip from sdcard,locate the root file where you have copied it (Internal storage or sdcard) and select it.
8-flash the root file.
9-reboot the tab,you will see a new app called SuperSU or Superuser,and surprise your tab is rooted
Credits goes to the developers who made the rooting method possibe on this device,i didn't make them so im only here to share information.
Have a nice day!!