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

There are \( \prod_{i=1}^{n} (\alpha_i + 1) \) elements \( \beta \) that satisfy the condition.

Step-by-step solution

Recognize the Condition

You are given a vector \( \alpha = (\alpha_1, \ldots, \alpha_n) \) where each \( \alpha_i \) is a natural number. You need to find all vectors \( \beta = (\beta_1, \ldots, \beta_n) \) such that each \( \beta_i \) is less than or equal to \( \alpha_i \).

Count Possibilities for Each Dimension

For each \( i \), \( \beta_i \) can be any natural number from 0 up to \( \alpha_i \). Therefore, for each \( i \), \( \beta_i \) has \( \alpha_i + 1 \) possible values.

Calculate Total Combinations

Since the choices for each \( \beta_i \) are independent, multiply the number of possibilities for each \( i \). The total number of elements is given by the product \( (\alpha_1 + 1) \times (\alpha_2 + 1) \times \ldots \times (\alpha_n + 1) \).

Conclusion

The total number of vectors \( \beta = (\beta_1, \ldots, \beta_n) \) that satisfy the conditions is \( \prod_{i=1}^{n} (\alpha_i + 1) \).

Key concepts

Vector spaces

In mathematics, a vector space is a collection of objects called vectors. These vectors can be added together and multiplied by scalars, which are typically real numbers. Vector spaces are fundamental in linear algebra and provide the framework for dealing with vectors in a systematic way. They can represent various quantities and concepts in physics and engineering,
and even extend to functions and polynomials. In our problem, we are dealing with vectors in a specific space composed of natural numbers. Such vectors are represented in the form \( \beta = (\beta_1, \ldots, \beta_n) \), where each component \( \beta_i \) is an element of the natural numbers.

• Each \( \alpha_i \) and \( \beta_i \) acts as a constraint, indicating a bound within which the vectors operate.
• The concept of vector spaces helps us understand how the constraints apply to each vector component independently,
  leading to the solution by evaluating each dimension separately.

Natural numbers

Natural numbers are the set of positive integers starting from 1, i.e., \( \mathbb{N} = \{1, 2, 3, \ldots \} \). These numbers are foundational in counting and ordering, and they serve as the building blocks for defining other mathematical structures and concepts like
integers and real numbers. In the given combinatorics problem, both \( \alpha \) and \( \beta \) are elements of \( \mathbb{N}^n \) where each \( \alpha_i \) and \( \beta_i \) are natural numbers.

• Each vector component \( \beta_i \) must satisfy \( 0 \leq \beta_i \leq \alpha_i \), meaning \( \beta_i \)
  can be zero or any positive natural number up to \( \alpha_i \).
• This requirement stems from the characteristics of natural numbers and allows for a defined and countable number of vector combinations,
  as calculated in the solution.

Cartesian product

The Cartesian product is a fundamental concept in set theory that represents all possible pairs of elements taken from two sets. When we extend this to more than two sets, we get an n-dimensional array representing all possible combinations across multiple sets.
In mathematical terms, if you have sets \( A \) and \( B \), the Cartesian product \( A \times B \) is the set of all pairs \((a, b)\) where \( a \in A \) and \( b \in B \). In our problem, the vectors \( \beta \) are essentially coordinates in an n-dimensional Cartesian product space, bounded by the vector \( \alpha \).

• These combinations are independent of one another, meaning each dimension \( \beta_i \) acts independently,
  analogous to different axes in a Cartesian coordinate system.
• This concept aids in understanding how the combination of values across different components
  results in the total number of vector permutations, calculated as a product of possibilities in each dimension,
  \( (\alpha_1 + 1) \times (\alpha_2 + 1) \times \ldots \times (\alpha_n + 1) \).

Problem solving

Problem-solving in combinatorics involves finding effective strategies to count arrangements or selections within specific conditions. In this exercise, you are tasked with counting possible combinations of vector elements \( \beta_i \). The process breaks down into steps:

• First, recognize any given conditions that constrain your options, in this case, each \( \beta_i \leq \alpha_i \).
• Count the possible values each component can take. Since each \( \beta_i \) ranges from 0 to \( \alpha_i \), that gives \( \alpha_i + 1 \) total options per dimension.
• To find the overall number of combinations, multiply the number of options for each dimension, reflecting the independence
  in the selection of each component.
• Finally, understanding the conceptual basis for these calculations, such as vector spaces and Cartesian products,
  enhances problem-solving skills by linking abstract mathematical concepts to concrete problems.