Question

How many ways can six candy bars be distributed among three children if every child is to receive at least one candy bar?

Short answer

There are 10 ways to distribute the six candy bars among the children ensuring each child gets at least one candy bar.

Step-by-step solution

Understand the Problem

We need to determine the number of ways to distribute six candy bars among three children, ensuring that each child receives at least one candy bar.

Initial Allocation

Start by giving each child one candy bar. This is necessary to ensure that each child gets at least one, which uses up 3 candy bars.

Calculate Remaining Candy Bars

After each child receives one candy bar, there are 6 - 3 = 3 candy bars left to distribute freely among the three children.

Apply Stars and Bars Method

We treat the problem as placing the remaining 3 candy bars (stars) into 3 indistinguishable groups (children). According to the stars and bars method, the number of ways to distribute these candy bars is given by the formula: \( \binom{n + k - 1}{k - 1} \), where \( n \) is the number of items to place and \( k \) is the number of groups.

Calculate Combinations

Substitute \( n = 3 \) and \( k = 3 \) into the formula: \( \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} \).

Compute the Binomial Coefficient

Calculate \( \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \). Thus, there are 10 ways to distribute the remaining candy bars.

Key concepts

Stars and Bars Method

The Stars and Bars Method is a popular technique used in combinatorics to solve problems related to distributing identical items into distinct groups. Imagine you have to distribute a number of identical candy bars among children. The stars (⭐) represent the items to be distributed, and the bars (|) divide those items into different groups for the children.

This method becomes particularly handy when dealing with questions about unrestricted distributions. For example, after initially giving each child one candy bar, we are left with distributing the remaining ones. Here, the key is to determine the arrangement of stars and bars.

To use this technique, remember:

• Stars: Represent the items you are distributing.
• Bars: Indicate divisions between groups.

When distributing the remaining 3 candy bars among 3 children, we set up the problem as arranging 3 stars and 2 bars (since if there are 3 children, you need 2 bars to section off the distribution). This is where the formula for combinations comes into play, helping us calculate the possible arrangements.

Binomial Coefficient

The Binomial Coefficient is a crucial mathematical tool in combinatorics used to compute the number of ways to choose a subset of items from a larger set, without considering the order. It's denoted as \( \binom{n}{k} \), representing the number of combinations of \( n \) items taken \( k \) at a time.

In our problem, after initially distributing one candy bar to each child, we used the binomial coefficient to determine the number of ways to distribute the last 3 candy bars. The expression for this is \( \binom{5}{2} \), which gives us the number of combinations to arrange 5 objects (3 stars + 2 bars) with 2 "bars" dividing them.

To calculate a binomial coefficient, apply the formula:

\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

In our exercise, it simplified to:

\[ \binom{5}{2} = \frac{5!}{2!3!} = 10 \]

This means there are 10 unique ways to distribute the remaining candy bars, ensuring each child gets at least one.

Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. The concept is essential in computer science and combinatorics, especially when dealing with integers, graphs, and logical statements.

In combinatorics, one of its branches, we often study how to count, arrange, and structure objects. Problems such as the distribution of candy bars among children, where objects are distinct and countable, are typical of discrete mathematics. The stars and bars technique and the binomial coefficient are both tools frequently used in solving such type of problems.

Here are key areas linked to discrete mathematics:

• Counting: Focuses on the enumeration of objects under defined criteria, just like distributing candy bars.
• Structures: Studies how things fit together in discrete settings, such as groups and sets.

Understanding these fundamental concepts allows us to approach and solve a wide variety of problems within discrete realms, such as our original candy bar problem, in a systematic and logical manner.