Cannot use mkdir in home directory: permission denied (Linux Lubuntu)

you can try writing the command using 'sudo':

sudo mkdir DirName

As @kirbyfan64sos notes in a comment, /home is NOT your home directory (a.k.a. home folder):

The fact that /home is an absolute, literal path that has no user-specific component provides a clue.

While /home happens to be the parent directory of all user-specific home directories on Linux-based systems, you shouldn't even rely on that, given that this differs across platforms: for instance, the equivalent directory on macOS is /Users.

What all Unix platforms DO have in common are the following ways to navigate to / refer to your home directory:

• Using cd with NO argument changes to your home dir., i.e., makes your home dir. the working directory.
  • e.g.: cd # changes to home dir; e.g., '/home/jdoe'

• Unquoted ~ by itself / unquoted ~/ at the start of a path string represents your home dir. / a path starting at your home dir.; this is referred to as tilde expansion (see man bash)
  • e.g.: echo ~ # outputs, e.g., '/home/jdoe'

• $HOME - as part of either unquoted or preferably a double-quoted string - refers to your home dir. HOME is a predefined, user-specific environment variable:
  • e.g.: cd "$HOME/tmp" # changes to your personal folder for temp. files

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Thus, to create the desired folder, you could use:

mkdir "$HOME/bin"  # same as: mkdir ~/bin

Note that most locations outside your home dir. require superuser (root user) privileges in order to create files or directories - that's why you ran into the Permission denied error.