Question

Use multisets to determine the number of ways to pass out \(k\) identical apples to \(n\) children. Assume that a child may get more than one apple.

Short answer

There are \(\binom{k+n-1}{n-1}\) ways to distribute the apples.

Step-by-step solution

Understand the Multiset and Problem Context

To solve this problem using multisets, understand that distributing apples can be translated into finding the number of non-negative integer solutions to the equation \(x_1 + x_2 + ... + x_n = k\). Here, \(x_i\) represents the number of apples received by the \(i^{th}\) child.

Setup the Equation for Distribution

The task is to find the integer solutions to the equation given by \(x_1 + x_2 + ... + x_n = k\), where each \(x_i\) (the apples a child gets) is a non-negative integer, \(x_i \geq 0\).

Utilize the 'Stars and Bars' Theorem

The 'Stars and Bars' theorem states that the number of solutions to the equation \(x_1 + x_2 + ... + x_n = k\) in non-negative integers is given by \(\binom{k+n-1}{n-1}\). Here, 'stars' represent apples and 'bars' are dividers between groups of apples for each child.

Calculate Using the Stars and Bars Formula

Now, insert the values into the formula where \(k\) is the number of apples and \(n\) is the number of children. The solution is the combinatorial expression \(\binom{k+n-1}{n-1}\). For instance, if \(k=5\) (apples) and \(n=3\) (children), calculate \(\binom{5+3-1}{3-1} = \binom{7}{2}\).

Compute the Combinatorial Expression

Calculate the binomial coefficient \(\binom{7}{2}\) using the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\). So, \(\binom{7}{2} = \frac{7!}{2!(5)!} = \frac{7 \times 6}{2 \times 1} = 21\).

Key concepts

Stars and Bars Theorem

The 'Stars and Bars' theorem is a handy tool in combinatorics, especially when distributing identical items, like apples, among distinct groups, like children. Imagine you have a certain number of identical items (our 'stars') that you need to distribute into different categories or bins (our 'bars').
The theorem helps us find the number of ways to distribute these items such that each category or bin could potentially get zero or more items.
To apply the 'Stars and Bars' method to our problem, imagine the apples as stars lined up in a row, and the bars as dividers between groups of stars. The question then becomes: how can we place the bars to divide the stars into groups corresponding to the children?
Here is how the distribution works:

• Each star corresponds to an apple given to a child.
• Each bar separates apples given to different children.

This arrangement shows that with the sum of stars, plus the number of bars placed, we arrive at the solution of distributing the apples efficiently.

Integer Solutions

When solving problems using the 'Stars and Bars' theorem, the focus is on finding integer solutions for equations. In our scenario, we need to determine how many ways we can assign whole numbers of apples to each child, with the total adding up to a specified number.
The equation in play here is:\[x_1 + x_2 + \ldots + x_n = k\]where:

• \( x_i \) represents the apples a specific child receives.
• \( k \) is the total number of identical apples available.
• \( n \) is the number of children.

Each solution involves whole numbers only, thus referring to 'integer solutions'. The non-negative solutions signify that each child can receive zero or more apples, provided all apples are distributed.

Combinatorial Expressions

Combinatorial expressions are vital for solving distribution problems in combinatorics. They are algebraic forms that capture the essence of counting problems involving arrangements or selections.
In our context, combinatorial expressions help express the complexity of distributing items (apples) into groups (children) as a mathematical equation.
The key expression from the 'Stars and Bars' theorem is given by:\[\binom{k+n-1}{n-1}\]This formula specifies the number of distinct ways to achieve our desired distribution:

• The numerator \( k+n-1 \) represents the total initial set of positions available for stars and bars combined.
• The denominator \( n-1 \) indicates the positions we choose for placing the bars among the stars.

This equation efficiently calculates all possible distributions with just a simple expression.

Binomial Coefficient

The binomial coefficient is a central concept in combinatorics, often used to find combinations of items from a larger set. It's represented as \( \binom{n}{r} \), denoting the number of ways to choose \( r \) items from \( n \) items without regard to order.
Mathematically, the binomial coefficient is expressed as:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Here’s how it applies in our problem:

• \( n! \) (n factorial) means multiplying all whole numbers from 1 up to \( n \).
• The formula calculates the number of combinations by dividing \( n! \) by the product of \( r! \) and \((n-r)!\).
• The binomial coefficient ties directly into the 'Stars and Bars' theorem to compute potential distributions of apples.

By using these coefficients, you can solve not just this problem, but a host of other combinatorial distribution challenges effortlessly.