Permutations and Combinations for Samples with Limited Repetitions _ 1 PDH

Introduction This Python script addresses a fundamental problem in combinatorics: calculating the number of distinct permutations and combinations for a multiset (a set with duplicate elements). The trial is to determine how many ways we can select and arrange m items from a collection where individual elements have limited available quantities. A classic example is finding all possible 2-letter words from the letters in "aaabbbbc". The script efficiently solves this by leveraging the mathematical principles of generating functions and the itertools module to iterate through all possible selection scenarios, validating them against the constraints, and applying combinatorial formulas to count the valid outcomes. Learning Objectives Calculate Valid Distributions: For a multiset (e.g., 3 'a's, 4 'b's, 1 'c'), determine all the distinct ways to distribute m selections across the different element types. For example, for m=2, the valid distributions are (2,0,0), (0,2,0), (1,1,0), (1,0,1), (0,1,1). Count Combinations: The number of valid distributions directly equals the number of unique combinations (where order does not matter). The script counts each valid distribution as one combination. Count Permutations: For each valid distribution of elements (e.g., one 'a' and one 'b'), calculate the number of distinct sequences (permutations) that can be formed. This is done using the multinomial coefficient formula: m! / (x1! * x2! * ... * xk!), where x1, x2, ... xk are the counts of each element in the distribution. Offer a Solution: Accept user input for the number of distinct elements, their respective frequencies, and the selection size m to solve a wide range of similar problems beyond the initial example. PDH Eligibility To request a PDH certificate, don't hesitate to email author with solved quiz. Disclaimer The author and publisher of this book have made effort to ensure the accuracy of the information contained herein. However, this book is shared for informational purposes only and is not intended as professional, legal, medical, or financial advice. The reader is responsible for their own actions and decisions. The author shall not be held liable or responsible for any loss, damage, or injury caused, or alleged to be caused, directly or indirectly, by the information contained in this book.