Question

How many monomials in variables \(x_1, \ldots, x_n\) have total degree \(m\) ?

Short answer

The number of monomials is given by \(\binom{m+n-1}{n-1}\).

Step-by-step solution

Understanding the Problem

We need to count how many ways we can form a monomial of degree \(m\) using \(n\) variables \(x_1, x_2, \ldots, x_n\). Each monomial has the form \(x_1^{a_1} x_2^{a_2} \ldots x_n^{a_n}\), where the sum \(a_1 + a_2 + \cdots + a_n = m\).

Introducing the Stars and Bars Method

This is a problem of finding the number of non-negative integer solutions to the equation \(a_1 + a_2 + \cdots + a_n = m\). The Stars and Bars method provides a way to solve this by conceptualizing it as distributing \(m\) indistinguishable objects (stars) into \(n\) distinguishable boxes (variables), separated by \(n-1\) bars.

Using Stars and Bars Formula

According to the Stars and Bars theorem, the number of ways to distribute \(m\) indistinguishable objects into \(n\) distinguishable boxes is given by the binomial coefficient \(\binom{m+n-1}{n-1}\). This is calculated as \(\frac{(m+n-1)!}{m! (n-1)!}\).

Solving the Problem

To find the number of monomials of degree \(m\) in \(n\) variables, we calculate \(\binom{m+n-1}{n-1}\) using the formula derived from the Stars and Bars method.

Example Calculation

For instance, if \(n=3\) and \(m=4\), calculate \(\binom{4+3-1}{3-1} = \binom{6}{2} = 15\). So, there are 15 different monomials that can be formed.

Key concepts

Monomials in Algebra

A monomial in algebra is an expression that consists of a single term, represented as a product of constants and variables. The structure of a monomial includes a coefficient (which may be 1) and one or more variables raised to non-negative integer exponents. For example, in the monomial

• x^3y^2

"x" and "y" are variables, and 3 and 2 are the exponents. Each of these variables can have different powers, but the defining feature is that all the terms are multiplied together without addition or subtraction.
The degree of a monomial is the sum of the exponents of its variables. If you have a monomial like

• x^3y^2z

its degree would be 6, as it is the sum of the exponents 3, 2, and 1.
Monomials are fundamental in algebra because they make up polynomials, which are more complex expressions consisting of multiple monomials connected by addition or subtraction.

Stars and Bars Method

The Stars and Bars method is a mathematical technique used in combinatorics to solve problems related to distributing identical items into different categories. It’s particularly useful when you want to find the number of solutions to an equation of the form: \[a_1 + a_2 + \dots + a_n = m\]where each \(a_i\) is a non-negative integer. In the context of algebra, the Stars and Bars method helps to figure out how many ways you can arrange a certain number of variables (stars) by using separators (bars) to indicate where one group ends and another begins.

• The stars represent the total degree you want, basically the sum of the exponents in the monomial.
• Bars are used to separate these stars into different groups, each corresponding to a variable in the monomial.

The theory states that if you have \(m\) stars and \(n\) variables, you need \(m + n - 1\) positions in total, made up of \(m\) stars and \(n - 1\) bars.
The calculation then becomes figuring out how many ways you can choose \(n - 1\) positions for bars out of the total \(m + n - 1\). This is where the binomial coefficient comes in handy.

Binomial Coefficient

The binomial coefficient is a key mathematical tool in combinatorics, symbolized as \(\binom{n}{k}\). It represents the number of ways you can choose \(k\) elements from a set of \(n\) elements without considering the order of selection. It is calculated using the formula:\[\binom{n}{k} = \frac{n!}{k! (n-k)!}\]where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
In algebra, binomial coefficients are used to expand binomials raised to a power. However, in the context of the Stars and Bars method, the binomial coefficient helps determine the number of ways to distribute exponents over multiple variables.

• For a problem where you need to distribute \(m\) items into \(n\) groups, it’s the number of ways \(n - 1\) dividers can be arranged among \(m + n - 1\) positions.
• This provides the number of possible monomials of a given degree across multiple variables.

Using the example of calculating different monomials, if you have 3 variables and want a total degree of 4, you compute it as \(\binom{6}{2} = 15\), showing there are 15 ways to distribute the degrees across the variables.