Question

Prove that in any class of 35 students, at least seven receive the same final grade, where the scale is \(\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{D}-\mathrm{F}\).

Short answer

By the Pigeonhole Principle, at least one grade must be assigned to at least 7 students.

Step-by-step solution

Understand the Problem

We need to prove that in a class of 35 students, if each has received a grade on a 5-grade scale (A, B, C, D, F), then at least one grade must be given to at least 7 students.

Define the Pigeonhole Principle

The Pigeonhole Principle states that if you place more items into fewer containers than there are items, at least one container must contain more than one item. In more formal terms, if 'n' items are put into 'm' containers, with n > m, at least one container must contain more than one item.

Identify the Variables

Here, the 'items' are the 35 students, and the 'containers' are the 5 possible grades: A, B, C, D, and F. Thus, in this context, we have n = 35 students and m = 5 grades.

Apply the Pigeonhole Principle

For the grade distribution in our problem, if the maximum number of students per grade were 6, each would account for 6 students, totaling only 30 students (6 students x 5 grades = 30 students). But we have 35 students, meaning at least one grade must accommodate more than 6 students.

Conclude by Calculation

If each grade had only 6 students, that accounts for 30 students max. Therefore, the remaining 35 - 30 = 5 students must receive grades as well. Hence, at least one particular grade is assigned to at least 7 students.

Key concepts

Discrete Mathematics

Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separate values. It contrasts with continuous mathematics, which deals with smoothly varying quantities.

In our problem, discrete mathematics plays a role in clearly defining separate entities — students receiving specific grades — rather than a continuous range of scores. Each student receives a discrete grade on the five-point scale: A, B, C, D, or F. This distinction is crucial as it forms the basis for applying the Pigeonhole Principle, ensuring that we are counting whole units rather than averaging or estimating.

Discrete mathematics values simplicity and clarity in proof construction, like ensuring that every number and category used is distinctly identified and managed. This makes it an excellent foundation for understanding logical puzzles and quantitative proof techniques, such as the one demonstrated in our grade assignment problem.

Combinatorics

Combinatorics is a field of mathematics concerned with counting, arrangement, and combination of objects. It is particularly useful when dealing with quantifiable events that adhere to certain constraints or limitations.

In the context of our exercise, combinatorics helps us handle the task of determining the distribution of grades among the 35 students. By considering all possible arrangements of these students across the five different grades, we effectively use combinatorial principles to understand how the Pigeonhole Principle forces certain distributions.

This is more than just counting: it's about understanding the inherent arrangements that come with given constraints. By knowing the total number of students and the number of grading bins (five in our exercise), we can predict that exceeding a certain threshold in any arrangement, such as having at least seven students per grade, is inevitable given the restrictions of the problem.

Proof Techniques

Proof techniques in mathematics are methods used to establish the truth of statements. They serve as the backbone for validating theories and propositions through logical reasoning. The pigeonhole proof used in the exercise offers a straightforward method for demonstrating a result that might initially seem counterintuitive.

In this problem, applying the Pigeonhole Principle is a proof by contradiction and enumeration. We start by assuming the maximal distribution that seems plausible under given constraints if each grade was to have a maximum of six students, which only covers 30 students out of 35.

The surplus of students — five in this case — forms the crux of the proof that forces at least seven students to share the same grade. This technique helps students appreciate how seemingly paradoxical outcomes can be logically predetermined using quantitative logic. The step-by-step breakdown is an excellent way to ensure all premises are accounted for and that the conclusion follows logically from the facts.