Guía de resolución de problemas de matemáticas combinatorias

Introduction

Problem solving is a crucial skill for university students who need it to succeed in their academic and professional careers. One area of mathematics where problem-solving skills are highly sought after is combinatorial math, which deals with counting techniques. In this article, we will present a comprehensive problem-solving guide for four fundamental concepts in combinatorial math: the principle of the sum, the principle of the multiplication, permutations, and combinations. We will discuss these concepts using real-life examples and practical exercises. Let's begin!

Principle of the Sum

The principle of the sum is used when we want to find the number of favourable outcomes from several alternative possibilities. It states that if there are n ways of doing something one way, and m ways of doing it another, then there are n + m ways of doing it either way. For example, if you have two shirts - blue and red - and two pairs of pants - jeans and khakis - you can choose from a total of 4 outfits (blue + jeans, blue + khakis, red + jeans, and red + khakis).

Exercise 1.1

You have three choices for breakfast, lunch, and dinner each day. How many different meal plans can you create over five days?

A) 125 B) 90 C) 30 D) 243

Hint: Apply the principle of the sum.

Principle of Multiplication

The principle of multiplication involves multiplying the number of ways together rather than adding them. This concept comes into play when considering repeating opportunities. A simple example would involve choosing 2 books from 4 available ones. When dealing with multiple categories, such as choosing two books from category A and two books from category B, the principle becomes 2 × 2 = 4 combinations.

Exercise 2.1

Suppose you have two suitcases, and each contains two sets of clothes. How many possible combinations of clothes could you wear if you select items from only one suitcase? Choose 2 items from the first suitcase and 2 from the second.

A) 16 B) 256 C) 1024 D) 4

Hint: Consider the principle of multiplication.

Permutations (Simple)

Permutations count the arrangements of objects without regard to their order. If there are n things to arrange, then there are n ways of arranging them according to the factorial method, i.e., n! ways. For a more in-depth understanding of permutations, consider n = 4; it means there are 4! = 4 × 3 × 2 × 1 = 24 ways to arrange them.

Exercise 3.1

If there are 4 options for each of the following actions, how many possible combinations can be formed by selecting one option from each action?

A) 120 B) 720 C) 1440 D) 3240

Hint: Think about the concept of permutations.

Permutations (With Repetitions)

In scenarios involving repetition, we must count all unique permutations. To do so, we use the formula (n+r-1)! / r!(n-1)!, where n represents the size of the set, and r denotes the number of items chosen at a time.

Exercise 4.1

How many possible 2-digit numbers can be formed by using digits 0 through 11 with repetition allowed?

A) 1 B) 11 C) 1001 D) 11001

Hint: Take into account permutations with repetition.

Permutations (Circular)

When objects are considered circularly arranged, an additional permutation exists due to rotation. Thus, there are 2n! circular permutations.

Exercise 5.1

If there are 6 vowels (including space) in the English alphabet, how many possible circular permutations can be generated by rearranging them?

A) 7! B) 2.7! C) 6! D) 6! / 2! * 2

Hint: Remember that there are 2n! circular permutations.

Combinations (Simple)

Combinations are similar to permutations, except they do not consider the order of the elements. They are used when the objects are indistinguishable. The formula for finding the number of combinations is nCr = n! / (n-r)!r!.

Exercise 6.1

There are 5 people in the class. How many possible groups of 3 people can be formed?

A) 10 B) 20 C) 25 D) 35

Hint: Consider the concept of combinations.