The short answer : Shift + Win + S.

This will launch the Windows built-in snipping function, but not exactly the snipping tool.  Windows 10 Creators Update is required, which should be already installed about 4 years ago.

Before this, I have tried to figure out an alternate short cut because the print screen button is not working.  Windows 10 comes with a snipping tool.  But launching it every time is not too convenient.  Now, I am surprised there is such a handy hot key to capture part of the screen directly.

There is also a way to customize the hot key for the snipping tool actually.  An extract from superuser.com forum below:

  1. Find your Snipping Tool shortcut (what you use to start the Snipping Tool manually). If you can not find the shortcut, you can make a shortcut from SnippingTool.exe (in theSystem32 folder).
  2. Right-click the shortcut and select properties.
  3. Select the Shortcut tab.
  4. Click inside the Shortcut key box, then hit the hotkey(s) you want. Note that you can not type the key you want.

More information can be found here : Is there a way to snip part of the screen with a hotkey?

Anyway, advanced users may just want to use a paid solution like Snagit.