.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/combinatorics/counting/plot_01_poker_hands.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_01_poker_hands.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_combinatorics_counting_plot_01_poker_hands.py:


Counting poker hands
=========================

Counts 5-card poker hands and a few hand types, and cross-checks the
count against brute-force enumeration for a smaller deck.

.. GENERATED FROM PYTHON SOURCE LINES 10-12

.. code-block:: Python

    from mathematicskit.combinatorics import combinations_count, generate_combinations, multinomial_coefficient








.. GENERATED FROM PYTHON SOURCE LINES 13-15

Total 5-card hands from a standard 52-card deck
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 15-19

.. code-block:: Python


    total_hands = combinations_count(52, 5)
    print(f"total 5-card hands: {total_hands}")





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

 .. code-block:: none

    total 5-card hands: 2598960




.. GENERATED FROM PYTHON SOURCE LINES 20-22

Number of ways to deal a full house (3 of one rank, 2 of another)
-------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-26

.. code-block:: Python


    full_houses = combinations_count(13, 1) * combinations_count(4, 3) * combinations_count(12, 1) * combinations_count(4, 2)
    print(f"full houses: {full_houses} (probability {full_houses / total_hands:.6f})")





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

 .. code-block:: none

    full houses: 3744 (probability 0.001441)




.. GENERATED FROM PYTHON SOURCE LINES 27-29

Cross-check via brute-force enumeration on a small 8-card deck
-------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-34

.. code-block:: Python


    small_deck = list(range(8))
    hands = generate_combinations(small_deck, r=3)
    print(f"\nC(8,3) = {combinations_count(8, 3)}, generated {len(hands)} hands")





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

 .. code-block:: none


    C(8,3) = 56, generated 56 hands




.. GENERATED FROM PYTHON SOURCE LINES 35-37

Multinomial coefficient: ways to deal a 13-card deck into 4 equal hands
------------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-40

.. code-block:: Python


    deals = multinomial_coefficient(12, [3, 3, 3, 3])
    print(f"ways to split 12 cards into 4 groups of 3: {deals}")




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

 .. code-block:: none

    ways to split 12 cards into 4 groups of 3: 369600





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_combinatorics_counting_plot_01_poker_hands.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_poker_hands.ipynb <plot_01_poker_hands.ipynb>`

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

      :download:`Download Python source code: plot_01_poker_hands.py <plot_01_poker_hands.py>`

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

      :download:`Download zipped: plot_01_poker_hands.zip <plot_01_poker_hands.zip>`


.. only:: html

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

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