Question

Prove that in any class of more than 101 students, at least two must receive the same grade for an exam with grading scale of 0 to 100 .

Short answer

At least two students must receive the same grade because there are more students than grades.

Step-by-step solution

Define the Grading Scale and Students Count

The grading scale for the exam ranges from 0 to 100, inclusive. This means there are 101 possible grades: \(0, 1, 2, \ldots, 100\). Let there be \(n\) students in the class, where \(n > 101\).

Understand the Pigeonhole Principle

The Pigeonhole Principle states that if \(n\) items are distributed into \(m\) containers, and \(n > m\), then at least one container must contain more than one item. This principle will be applied to the distribution of grades among students.

Apply the Pigeonhole Principle

Imagine each possible grade (from 0 to 100) as a 'container' for the students' grades. Since there are 101 possible grades, and more than 101 students (i.e., \(n > 101\)), the Pigeonhole Principle tells us that at least one grade must be assigned to more than one student.

Conclude the Proof

Thus, since there are only 101 possible grades but more than 101 students, at least two students must share the same grade. This completes the proof by the application of the Pigeonhole Principle.

Key concepts

Discrete Mathematics

Discrete mathematics is a crucial field that deals with distinct and separate values. Unlike continuous mathematics, which deals with smooth and flowing values, discrete mathematics focuses on countable, distinct values.
This branch of mathematics is foundational for computer science, logic, and more. It includes concepts like integers, graphs, and logic statements, all of which are key to developing algorithms.

• Integers and Natural Numbers: Discrete math often uses whole numbers and their properties.
• Graph Theory: A major area dealing with nodes and connections, useful for network analysis.
• Logic: Essential for formulating proofs and reasoning.

In the context of our problem, discrete math helps us understand how to count and distribute grades, employing logical principles to conclude that some grades must repeat.

Combinatorics

Combinatorics is the study of counting, arrangement, and combination of objects. It involves finding efficient ways to count, often without actually counting the entire set. This can be integral when dealing with complex systems.
One of the foundational tools in combinatorics is the Pigeonhole Principle, which is central to solving our exercise.

• Pigeonhole Principle: A powerful tool stating that if you have more items than containers, some containers must hold more than one item.
• Permutations and Combinations: Methods for arranging and selecting items, vital for determining possibilities.
• Factorials: Often used in calculations involving permutations.

In the proof given, combinatorics helps us see that distributing more students than available grades naturally leads to some grades being shared.

Mathematical Proofs

Mathematical proofs are logical arguments that confirm the truth of a statement. Proofs involve a series of reasoning steps that lead from premises to a conclusion. Learning how to construct proofs is a fundamental aspect of mathematics.
In our exercise, we utilized a type of proof known as proof by contradiction, where we assume the opposite of what we want to prove, and show that this assumption leads to a contradiction or impossible situation.

• Direct Proof: Directly shows a statement is true using known facts and logical steps.
• Contradiction: Assumes the opposite of what needs proving, then shows this results in a contradiction.
• Induction: Proves a statement for all natural numbers by proving for the first, and assuming true for an arbitrary number.

Here, the proof used suggests that more students than grades must result in repeated grades, as shown with the Pigeonhole Principle, completing the logical argument.