Question

The number of ways of distributing 8 identical balls in 3 distinct boxes so that no box is empty is (a) 5 (b) \({ }^{8} \mathrm{C}_{3}\) (c) 38 (d) 21

Short answer

There are 21 ways to distribute 8 identical balls in 3 distinct boxes, ensuring that no box is empty. The correct answer is (d) 21. We used the "stars and bars" combinatorial method to calculate this by first putting one ball in each box and then arranging the remaining 5 balls and 2 bars to represent the distribution among the boxes. The number of ways to arrange the balls and bars was calculated using the formula \( \frac{(\text{total objects})!}{(\text{count of balls})!(\text{count of bars})!} \) and simplifying the expression.

Step-by-step solution

Understanding the "Stars and Bars" Method

The "stars and bars" method is a combinatorial technique used to count the number of ways to distribute objects (in this case, balls) among a certain number of containers (in this case, boxes). In this method, imagine the balls as stars, and we will put bars between them to form "partitions" that indicate the division of balls among the boxes. The method will help to ensure no box is empty.

Ensure No Box is Empty

Since no box can be empty, let's start by putting one ball in each box first. So, we are now left with 5 balls that need to be distributed among the 3 distinct boxes.

Arrange the remaining balls and bars

We have 5 balls left to distribute among the 3 boxes. To use the "stars and bars" method, we are going to place 2 bars (3 boxes - 1) among the 5 balls to make partitions representing each box. So, we have 5 balls (or stars) and 2 bars that we need to arrange in such a manner that the balls are separated by the bars. For example, the arrangement: * * * | * | * * represents box 1 with 3 balls, box 2 with 1 ball, and box 3 with 2 balls.

Calculate the number of ways to arrange balls and bars

Now, we need to calculate the number of ways to arrange these 5 balls and 2 bars. This can be found using the formula: \( \text{Ways to arrange} = \frac{(\text{total objects})!}{(\text{count of balls})!(\text{count of bars})!} \) Total objects in our case are 5 balls and 2 bars, so we have a total of 7 objects. Substituting the values: \( \text{Ways to arrange} = \frac{7!}{5!2!} \)

Simplify the expression

We simplify the expression to get the answer: \( \text{Ways to arrange} = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(5 \times 4 \times 3 \times 2 \times 1)(2 \times 1)} \) By canceling out the common terms, we get: \( \text{Ways to arrange} = \frac{7 \times 6}{2} = 7 \times 3 = 21\) So, there are 21 ways to distribute 8 identical balls in 3 distinct boxes, ensuring that no box is empty. The correct answer is (d) 21.

Key concepts

Stars and Bars Method

The stars and bars method is a popular technique in combinatorics, particularly useful for distributing identical items into distinct groups. Imagine you have some 'stars' (objects to distribute) and 'bars' (which are separators between different groups). This visual method helps determine all possible ways to distribute these stars across the groups.

For example, if you need to distribute 8 identical balls into 3 boxes, and each box needs at least one ball, you start by placing 1 ball in each box. Then you have 5 balls left to distribute freely among the boxes.

Visualizing this, you arrange the remaining balls and place bars to create separations between different sections, ensuring every box has its share. If you have 3 boxes, you require 2 bars to ensure separation. The problem then becomes arranging these 5 stars and 2 bars in a sequence, e.g., `* * * | * | * *`, which signifies how the balls are grouped into boxes.

By calculating the number of unique sequences of stars and bars, you determine the distribution ways. This method is particularly helpful in solving problems where the partitioning of objects is necessary or when bounds are set (like no box can be empty).

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics related to counting arrangements and selections.

**Permutations** are all the possible ways to arrange a set of items when the order does matter. For "arrange", think about creating a different sequence out of given items.

• For instance, for three balls labeled A, B, and C, the permutations would include ABC, ACB, BAC, BCA, CAB, and CBA, meaning there are 6 different ways to arrange these balls.

**Combinations**, on the other hand, concern scenarios where the order does not matter. It's about selecting items from a group, and the order of selection does not change the group.

• Using combinations, if you are choosing 2 balls from a set of 3 labeled A, B, C, you have combinations AB, AC, and BC, resulting in 3 different groups.

In our main example, the stars and bars method indirectly uses combinations by identifying different ways to group the stars with bars. This ensures that each unique set or combination of separations is counted correctly. The equation used involves factorials, which are key calculations in permutations and combinations, helping us figure out possible arrangements and selections.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Unlike calculus that deals with continuity, discrete math focuses on distinct, separate elements.

This field of mathematics encompasses a range of topics, including logic, set theory, graph theory, and, as relevant to our main problem, combinatorics. The countable scenarios of objects, such as how many different ways you can choose or arrange a number of elements, fall into this category.

• For instance, when you are determining how to distribute balls into boxes or when using stars and bars, you are practicing discrete mathematics. Here, the distinct nature is represented by the separate boxes and separations using bars.

Discrete math is critical in computer science, cryptography, network modeling, and algorithm design due to its focus on distinct and often finite systems. The applications are vast, extending beyond theoretical calculations into solving practical and complex problems found in various technological and scientific fields. It provides the tools necessary for analyzing the intricate and exact nature of real-world and theoretical constructs alike.