How can I root the Galaxy Tab 10.1? How can I root the Samsung GT-P7510? I've tried z4root and many others, but they all seem to support Android 3.1 and not 3.2. I gather 3.2 is specifically designed to eliminate the exploit that z4root and others depend on.
You will need to flash a pre-rooted 3.2 ROM onto your Tab.

A Google search brings these instructions for installing a custom Recovery:


  First, you will need Windows computer. If you don’t have one, borrow a
  Windows laptop from your friend or family member.
  
  Second, make sure you’ve installed Galaxy Tab 10.1 Windows drivers.
  (Download it here)
  
  Third, put your Galaxy Tab 10.1 into ODIN download mode by holding
  down both Power Button and Volume Down buttons. Right after the
  Samsung logo shows, you will see two menus, let go of both buttons
  immediately and then hit Volume Up button, you should be in ODIN
  download mode.
  
  Fourth, download the tabrooteasy.zip file and unzip:   Download
  tabrooteasy.zip
  
  Run the Odin3 v1.85.exe program, then choose PDA, choose
  “recovery-cwm_4.0.0.4-sam-tab-10.1.tar.md5″ and hit Start.
  
  You can reboot your Tab 10.1 by holding down the Power button.


The instructions go on to rooting Android 3.1, so you will need to slightly alter those.  Instead of using the "root" ZIP included in the package, you will need to obtain a pre-rooted Android 3.2 ROM.  There are many such ROMs available on XDA-Developers forum, so take your pick (e.g. this one,) download the ZIP file, and copy it to the SD Card.

Continue on with the instructions to reboot into the ClockworkMod Recovery:


  Reboot into ClockworkMod Recovery by holding down both Power Button
  and Volume Down buttons. Right after the Samsung logo shows, you will
  see two menus, let go of both buttons immediately and then hit Volume
  Down, then Volume Up button, you should be in ClockworkMod Recovery.


You will now need to perform a factory reset/data wipe to avoid any compatibility issues.  With the volume buttons for up/down, navigate to "wipe data/factory reset", and select it by pressing the Power button, and then scroll down to and select the "Yes" option.  Once that operation completes, you can continue with the instructions, substituting the file name they are suggesting with the one you've downloaded:


  ... choose “install zip from
  sdcard” (using Volume buttons) and hit the Power button.  Next, choose
  “choose zip from sdcard” and hit the Power button.  We are going to
  zip the Root/SU files again by going to “choose zip from sdcard” and
  hit the Power button.  Choose the [ insert file name of downloaded ROM here ]
  file and hit the Power button, the hit the “Yes” button.
  
  When that is done, you just need to reboot your Tab 10.1 and you
  should have a fully-rooted Android 3.2 Galaxy Tab 10.1.


According to some comments this method should work on the WiFi-only model of the Galaxy Tab 10.1 (GT-P7510).  However, you might encounter issues transferring the downloaded files onto the Tab.  A comment suggests a work-around:


  Ok folks. I got my Galaxy GT-P7510 wifi only tab (build HMJ37) rooted.
  Since I was not able to transfer the Samsung zip file to the root of
  the tab even tho Odin successfully installed the Clockworks recover
  file I decided to try a different approach. I transferred the Samsung
  zip file to the Alarms folder of the tab. I then shut down my tap and
  powered it back up in Clockworks recovery mode. When in recovery mode
  I selected “install zip from sdcard”. I then selected “choose zip from
  sdcard”. I selected the folder “Alarms” then selected the Samsung zip.
  I then selected “Yes – Install Samsung zip”. After the successful
  install I rebooted the tab, went to apps and BINGO…..SUPERUSER!!! I
  then successfully installed several apps that required a rooted tab.


Good luck!