Question

Prove that for any 44 people, at least four must be born in the same month.

Short answer

At least four people must be born in the same month.

Step-by-step solution

Problem Restatement

We need to demonstrate that among 44 people, there is at least one month in which at least four people share a birthdate. This is a typical application of the pigeonhole principle.

Identify the Pigeonholes and Pigeons

The 12 months of the year represent the pigeonholes, and the 44 people are the pigeons.

Apply the Pigeonhole Principle

According to the pigeonhole principle, if we have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. Mathematically, if you have more than \(n\times (k-1)\) pigeons and \(n\) pigeonholes, at least one pigeonhole must contain \(k\) or more pigeons.

Calculate Upper Bound for Pigeons per Month

For our case, we calculate: \(12\times 3 = 36\). With 44 pigeons (people) and 12 pigeonholes (months), since \(44 > 36\), by the pigeonhole principle, at least one month must have \((44 - 36) + 3 = 4\) or more people.

Conclusion

The calculation confirms that with 44 people distributed over 12 months, there must be at least one month with at least four people having a birthday. Therefore, the statement is proven.

Key concepts

Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are inherently discrete rather than continuous. Unlike calculus, which deals with continuous processes, discrete mathematics deals with distinct and separate values.

In the context of our exercise, discrete mathematics helps us explore scenarios involving a finite set of elements, like birthdays spread across different months. This field includes a variety of topics, such as graph theory, set theory, and combinatorics, which all focus on counting, arrangement, and the structure of sets and relationships.

A prime example of a discrete mathematics concept is the Pigeonhole Principle, which forms the basis of our original exercise. It's often used to demonstrate that certain outcomes are inevitable when distributing a finite set of items into a fixed number of categories. This kind of reasoning is typical in discrete mathematics, where problems often involve determining the optimal solution for complex arrangements without relying on continuous variables.

Mathematical Proofs

Mathematical proofs are logical arguments used to establish the truth of a mathematical statement. They are essential in discrete mathematics to ensure that conclusions drawn from mathematical models are reliable and accurate.

The exercise we looked at uses a form of proof by contradiction, with the Pigeonhole Principle as its basis. By attempting to distribute 44 individuals (pigeons) evenly across 12 months (pigeonholes), we demonstrate that at least one month will inevitably contain at least four people.

Proofs rely on clear structures and logical deductions that follow from assumptions. In our case, the assumption is that there are 12 months. The deduction is that with more people than what would allow for three people per month, certain months must have more than three birthdays, confirming our statement.

• Start with an observation or known principle (like the Pigeonhole Principle).
• Develop a logical sequence of statements based on this principle.
• Conclude from these statements to confirm the initial claim.

Combinatorics

Combinatorics is a field of mathematics focusing on counting, arranging, and understanding the structure of discrete objects. It involves techniques to count permutations, combinations, and other complex arrangements.

In our exercise, combinatorics comes into play with how we count and distribute people across months. By leveraging the Pigeonhole Principle, a tool that comes from combinatorial logic, we ensured proper distribution of people.

Using the principle helps us understand the minimal conditions needed for an outcome to be achieved. It effectively shows us how arrangements and orders play a critical role.

• Identify the number of items and categories.
• Use combinatorial principles to predict outcomes.
• Apply logical reasoning to arrive at conclusions about distributions.

These steps demonstrate how combinatorial thinking is vital in problem-solving, especially in discrete mathematics scenarios.