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Can I ask a fundamental question about PyCharm base interpreter?

I am trying to use Python3 with PyCharm.
So I installed Python(3.10.1) and PyCharm(2021.3.1)
But when I try to create a new project, PyCharm want me to choose 'base interpreter'
and I don't know which one to choose.
Here is the list of base interpreter:
I want to know what is difference between /usr/bin/python3 and Library/Frameworks/Python.framework/Versions/3.10/bin/python3

Usually the files under /usr/bin/ are symlinks to actual files, which means they are not real files but links to the actual file.
To see it for yourself go to the /usr/bin/ directory with a terminal and execute:
ls -a
I suppose macOS already ships with python and by installing Python 3.10 you added that last element to the list, which is also linked by /bin/usr/python3.
I'd go with /usr/bin/python3, more portable.

You can choose the one interpreter for the version of python you want. In your terminal, you can run python --version or python3 --version and it will display the version of python, Like this:
$ python --version
Python 2.7.16
$ python3 --version
Python 3.9.1
Performing which python, will give you the path to that python binary, like the following (although, this is where we are leaving "Python" and talking more about the OS and PATH.
$ which python3
/usr/local/bin/python3
For the other versions of Python you have installed, you can check their version by doing something like:
/Library/Frameworks/Python.framework/Versions/3.10/bin/python3 --version
You'll find that you may choose different versions of python for different projects. In the future, you may consider using venv to setup virtual python environments per project.
If you're just getting started with Python, I'd suggest sticking with the version you installed 3.10.1, and choosing that interpreter.

The base interpreter is the python.exe file that will run everything you do inside that project.
I would suggest you use Anaconda as a package manager and then create a virtual environment with the version of Python you need. Just follow these steps:
Getting started with Anaconda: https://docs.anaconda.com/anaconda/user-guide/tasks/pycharm/
Choosing an Anaconda virtual environment with Pycharm:
https://docs.anaconda.com/anaconda/user-guide/tasks/pycharm/
A package manager like Anaconda is very useful when working with Python.
Otherwise:
In your list you have many different versions of python. You should choose the base interpreter based on the python version you want to use.

Always select the latest version.
Since I use Python 3.9, it will be:
/usr/local/bin/python3.9
Based on your image, it would be:
/usr/local/bin/python3

The interpreter is what version your PyCharm project uses. It appears you have python2 and python3, so if you were writing in python2, you would use the python2 interpreter.
I would select the latest version, which in your case would be:
/usr/local/bin/python3

How install and use another version python(python 2.7) on linux with the default python version is python 2.6

There is a default python version, namely python 2.6, on the GPU server with Linux OS. Now I want to install a new python version on the server from its source, namely python 2.7. I should not change the default python version since I am not the administrator and some reason. So what should I do?

You can install your new version of Python. It should be accessible with the python27 command (which may be a symbolic link).
Then you will just have to check that the python symbolic link still points to python26.
Doing this, python will keep on execute Python 2.6 while python27 will execute Python 2.7

You can use virtualenv, to execute your programm in an environment with python 2.7.
Install virtualenv and virtualenvwrapper (for comfotable use.)
mkvirtualenv -p <your-python-version> would then start a virtual environment where the desired python version is the default.

To build on Tryph's answer, you can install that new version to your home directory, then in a directory specified within your PATH (like in .bash_profile), you can point to that directory and within there create a sym-link that points to the new python.
For instance, if you have a bin folder in your home directory that is specified in the path
ln -s /bin/python ~/bin/python

Python3 not found [duplicate]

I installed python3, I can open idle and it says it is running python3.0.1, but when I enter python3 in the terminal (on OSX) I get an error saying 'command not found'. Entering python gets me the 2.x version that came on the computer. Any advice on how I can access python3 from the terminal?
Thanks

First, don't use Python 3.0.1. It has many problems and was officially retired upon the release of Python 3.1 (currently 3.1.2). You can find the python.org Mac OS X installer for 3.1.2 here. Once it is installed, then you need to ensure that the bin directory from the 3.1.2 framework (/Library/Frameworks/Python.framework/Versions/3.1/bin) is on your shell search path. You can manually modify an appropriate shell startup file, like .bash_profile. Or just double-click the Update Shell Profile.command found in /Applications/Python 3.1. In either case, you will need to open a new terminal window or re-login. Another approach is to install Python 3.1 from MacPorts or another distributor. Also, alpha releases of Python 3.2 are now available from python.org and elsewhere.

Is it safe to set python bin in $PATH to another python version?

I have installed Anaconda3 just now, and I noticed that now, when I run python command from terminal, Python 3.5.1 |Anaconda 4.0.0 (64-bit)| is starting. Anaconda installer had added path to anaconda dir in $PATH and there is symlink from python to python3.5
My question is: will programs, that depends from python command and expects python2, work correctly, or I should remove symlink python from anaconda dir?

My question is: will programs, that depends from python command and
expects python2, work correctly?
Those programs should use full path of the python binary. Something like /usr/bin/python, and so $PATH is irrelevant. As long as you don't change /usr/bin/python, nothing will break.
If you remove the stuff that Anaconda has added, it's likely that Anaconda will not work properly.

That depends on your OS. Debian and Ubuntu both have ongoing projects to move the "default" version from 2 to 3 (also here). But it's not recommended to point /usr/bin/python to python3 if Python 2 is installed (see PEP 394).
If you want 'python' to be pointing to your 3.x install, you could use an alias (see here). This way you can use python in your session and at least don't change it on the whole system.

Installed python3, getting command not found error in terminal

I installed python3, I can open idle and it says it is running python3.0.1, but when I enter python3 in the terminal (on OSX) I get an error saying 'command not found'. Entering python gets me the 2.x version that came on the computer. Any advice on how I can access python3 from the terminal?
Thanks

First, don't use Python 3.0.1. It has many problems and was officially retired upon the release of Python 3.1 (currently 3.1.2). You can find the python.org Mac OS X installer for 3.1.2 here. Once it is installed, then you need to ensure that the bin directory from the 3.1.2 framework (/Library/Frameworks/Python.framework/Versions/3.1/bin) is on your shell search path. You can manually modify an appropriate shell startup file, like .bash_profile. Or just double-click the Update Shell Profile.command found in /Applications/Python 3.1. In either case, you will need to open a new terminal window or re-login. Another approach is to install Python 3.1 from MacPorts or another distributor. Also, alpha releases of Python 3.2 are now available from python.org and elsewhere.