.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/counting/plot_02_gersonides_formulas.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_combinatorics_counting_plot_02_gersonides_formulas.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_combinatorics_counting_plot_02_gersonides_formulas.py:


Gersonides' counting formulas
===================================

Levi ben Gershon's 1321 *Maaseh Hoshev* proved by induction that n
objects can be arranged in n! orders, that k of them can be arranged in
n!/(n-k)! ways, and that k can be chosen in n!/(k!(n-k)!) ways. This
example checks each formula against brute-force enumeration.

.. GENERATED FROM PYTHON SOURCE LINES 12-16

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.combinatorics import combinations_count, generate_combinations, generate_permutations, permutations_count








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Formula against enumeration
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-29

.. code-block:: Python


    items = "ABCDE"
    n = len(items)
    print(f"arrangements of {n} objects: formula {permutations_count(n)}, listed {len(generate_permutations(items))}")
    for k in range(n + 1):
        ordered = permutations_count(n, k)
        unordered = combinations_count(n, k)
        listed = len(generate_combinations(items, k))
        print(f"k = {k}: P(n,k) = {ordered:3d}, C(n,k) = {unordered:2d} (listed {listed}), P/C = k! = {ordered // unordered}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    arrangements of 5 objects: formula 120, listed 120
    k = 0: P(n,k) =   1, C(n,k) =  1 (listed 1), P/C = k! = 1
    k = 1: P(n,k) =   5, C(n,k) =  5 (listed 5), P/C = k! = 1
    k = 2: P(n,k) =  20, C(n,k) = 10 (listed 10), P/C = k! = 2
    k = 3: P(n,k) =  60, C(n,k) = 10 (listed 10), P/C = k! = 6
    k = 4: P(n,k) = 120, C(n,k) =  5 (listed 5), P/C = k! = 24
    k = 5: P(n,k) = 120, C(n,k) =  1 (listed 1), P/C = k! = 120




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Arrangements grow much faster than selections
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-41

.. code-block:: Python


    ks = range(0, 11)
    fig, ax = plt.subplots()
    ax.semilogy(list(ks), [permutations_count(10, k) for k in ks], "o-", label="ordered: P(10, k)")
    ax.semilogy(list(ks), [combinations_count(10, k) for k in ks], "s-", label="unordered: C(10, k)")
    ax.set_xlabel("k")
    ax.set_ylabel("count")
    ax.legend()
    ax.set_title("Choosing k of 10 objects")



.. image-sg:: /api/gallery/combinatorics/counting/images/sphx_glr_plot_02_gersonides_formulas_001.png
   :alt: Choosing k of 10 objects
   :srcset: /api/gallery/combinatorics/counting/images/sphx_glr_plot_02_gersonides_formulas_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Text(0.5, 1.0, 'Choosing k of 10 objects')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.049 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_counting_plot_02_gersonides_formulas.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_gersonides_formulas.ipynb <plot_02_gersonides_formulas.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_gersonides_formulas.py <plot_02_gersonides_formulas.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_gersonides_formulas.zip <plot_02_gersonides_formulas.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_