Let’s break down the statement and prove it step by step:

Statement: If n pigeons are assigned to m pigeonholes, and m < n, then at least one pigeonhole contains two or more pigeons.

To prove this, we’ll use a proof by contradiction. We’ll assume that all pigeonholes contain at most one pigeon and show that this leads to a contradiction.

Assumption: Each pigeonhole contains at most one pigeon.

Now, let’s analyze the situation. We have n pigeons and m pigeonholes, where m < n. If each pigeonhole contains at most one pigeon, the maximum number of pigeons we can distribute is equal to the number of pigeonholes, which is m.

Since m < n, it means we have more pigeons than the number of available pigeonholes. This implies that there are not enough pigeonholes to accommodate all the pigeons.

According to our assumption, each pigeonhole can hold at most one pigeon. Therefore, if we have more pigeons than pigeonholes, at least one pigeon will not have a pigeonhole to occupy.

This contradicts the initial assumption that all pigeons are assigned to pigeonholes. Thus, our assumption that each pigeonhole contains at most one pigeon must be false.

Therefore, by the principle of contradiction, we can conclude that if n pigeons are assigned to m pigeonholes, and m < n, then at least one pigeonhole contains two or more pigeons.

This principle is widely applicable in various mathematical problems and has many important applications in different fields, including number theory, graph theory, and computer science.

Statement: If n pigeons are assigned to m pigeonholes, and m < n, then at least one pigeonhole contains two or more pigeons.

To prove this, we’ll use a proof by contradiction. We’ll assume that all pigeonholes contain at most one pigeon and show that this leads to a contradiction.

Assumption: Each pigeonhole contains at most one pigeon.

Now, let’s analyze the situation. We have n pigeons and m pigeonholes, where m < n. If each pigeonhole contains at most one pigeon, the maximum number of pigeons we can distribute is equal to the number of pigeonholes, which is m.

Since m < n, it means we have more pigeons than the number of available pigeonholes. This implies that there are not enough pigeonholes to accommodate all the pigeons.

According to our assumption, each pigeonhole can hold at most one pigeon. Therefore, if we have more pigeons than pigeonholes, at least one pigeon will not have a pigeonhole to occupy.

This contradicts the initial assumption that all pigeons are assigned to pigeonholes. Thus, our assumption that each pigeonhole contains at most one pigeon must be false.

Therefore, by the principle of contradiction, we can conclude that if n pigeons are assigned to m pigeonholes, and m < n, then at least one pigeonhole contains two or more pigeons.

This principle is widely applicable in various mathematical problems and has many important applications in different fields, including number theory, graph theory, and computer science.