Question

Let \(n \in \mathbb{N}\) and \(\alpha=\left(\alpha_1, \ldots, \alpha_n\right) \in \mathbb{N}^n\). Determine the number of elements \(\beta=\left(\beta_1, \ldots, \beta_n\right) \in\) \(\mathbb{N}^n\) such that \(\beta_i \leq \alpha_i\) for \(1 \leq i \leq n\).

Short answer

The number of elements \( \beta \) is \( \prod_{i=1}^n (\alpha_i + 1) \).

Step-by-step solution

Understanding the Problem

We need to count how many sequences \( \beta = (\beta_1, \beta_2, \ldots, \beta_n) \) can be formed such that each \( \beta_i \) satisfies the condition \( \beta_i \leq \alpha_i \), where \( \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n) \).

Consider Each Component Individually

For each \( \beta_i \), there are \( \alpha_i + 1 \) possible values: \( 0, 1, 2, \ldots, \alpha_i \). This is because \( \beta_i \) can take any integer from 0 up to and including \( \alpha_i \).

Calculate Total Combinations

Since the components of \( \beta \) are independent, the total number of sequences \( \beta \) is the product of possibilities for each \( \beta_i \). Thus, the total number of such \( \beta \) is the product of \( (\alpha_i + 1) \) for all \( i \), which is \( (\alpha_1 + 1) \times (\alpha_2 + 1) \times \ldots \times (\alpha_n + 1) \).

General Formula

The formula for the number of elements \( \beta \) is \( \prod_{i=1}^n (\alpha_i + 1) \). This product represents multiplying \( \alpha_i + 1 \) possibilities for each of the \( n \) indices.

Key concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. They are often denoted as

• \( \mathbb{N} \), the symbol commonly used to represent them.

Natural numbers do not include negative numbers or fractions, making them your basic counting numbers that you start learning to count with in elementary math.
In many mathematical contexts, the number 0 is sometimes included in natural numbers, especially in computer science or some modern mathematical formulations.
In this exercise, we encounter natural numbers as both the context for the integer sequences and the upper bounds for the values allowed in the elements of these sequences.

Integer Sequences

An integer sequence is simply a list or collection of integers arranged in a specific order. The integers could be natural numbers, whole numbers, or even negative numbers, but in the scope of our problem, they remain within the realm of natural numbers.
When you hear the term 'sequence,' think of a series of numbers that you can follow along like steps. Each position in this sequence is indexed, typically numbered as 1, 2, 3, and so forth up to n, where each element corresponds to a specific index.

• In our exercise, we look at sequences \( \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n) \) and \( \beta = (\beta_1, \beta_2, \ldots, \beta_n) \), where each element of \( \beta \) has restrictions based on the corresponding \( \alpha \).
• Understanding sequences is crucial when calculating various mathematical problems, especially those related to combinatorics where order and arrangement are vital.

Counting Problems

Counting problems in combinatorics deal with determining the number of ways certain outcomes can occur. They require a systematic approach to ensure all possibilities are counted once and only once.
In our exercise, we're dealing with a classical counting problem where we need to determine how many possible sequences can be formed under given constraints.

• The key strategy is to look at each component \( \beta_i \) individually, understand its possible values, and then consider all components together as a whole.
• Here, each \( \beta_i \) can be any integer from 0 up to \( \alpha_i \), providing \( \alpha_i + 1 \) choices per element.
• To find the total count, you calculate the product \( (\alpha_1 + 1) \times (\alpha_2 + 1) \times \cdots \times (\alpha_n + 1) \). This principle is fundamental in combinatorics, ensuring you account for all combinations of choices for each position in a sequence.

This type of problem reinforces understanding of how foundational mathematical concepts interact, providing a comprehensive way to approach more complex problems.