Problem 25 - Discrete Mathematics with Applications

Question

The Bell numbers \(B_{n},\) named after the English mathematician Eric T. Bell (1883-1960) and used in combinatorics, are defined recursively as follows: $$B_{0}=1$$ $$B_{n}=\sum_{i=0}^{n-1}\left(\begin{array}{c} n-1 \\ i \end{array}\right) B_{i}, \quad n \geq 1$$ Compute each Bell number. $$B_{3}$$

Step-by-Step Solution

Answer

The Bell number for n = 3 is \(B_3 = 5\).

Step 1: Understanding the formula

The given formula for Bell numbers is as follows: \(B_n = \sum_{i=0}^{n-1} {n-1 \choose i} B_i \quad\) for \(n \geq 1\) Here, - \(B_n\) denotes the Bell number for the given value of n. - \(\sum_{i=0}^{n-1}\) is a summation from i=0 to i=n-1. - \({n-1 \choose i}\) are binomial coefficients, also expressed as \(\frac{(n-1)!}{i!(n-1-i)!}\), where "n! " represents the factorial of n. - \(B_i\) represents the Bell numbers for the values i=0 to i=n-1. Now that we understand the formula let's proceed to compute the Bell number for n=3.

Step 2: Plugging in the value of n

We need to compute the Bell number for \(n = 3\), i.e., \(B_3\). Using the formula: \(B_3 = \sum_{i=0}^{3-1} {3-1 \choose i} B_i\) This simplifies to: \(B_3 = \sum_{i=0}^{2} {2 \choose i} B_i\)

Step 3: Calculating each term in the summation

We will now calculate each term in the summation: 1. When i = 0: \({2 \choose 0} = \frac{2!}{(2-0)! 0!} = \frac{2}{2} = 1\) Since \(B_0 = 1\), we have: \({2 \choose 0} B_0 = 1 \cdot 1 = 1\) 2. When i = 1: \({2 \choose 1} = \frac{2!}{(2-1)!(1!)} = \frac{2}{1} = 2\) Since \(B_1 = 1\), we have: \({2 \choose 1} B_1 = 2 \cdot 1 = 2\) 3. When i = 2: \({2 \choose 2} = \frac{2!}{(2-2)! (2!)} = \frac{2}{2} = 1\) Since \(B_2 = 2\), we have: \({2 \choose 2} B_2= 1 \cdot 2 = 2\)

Step 4: Adding all computed terms to find B_3

Combining all calculated terms to find \(B_3\): \(B_3 = \sum_{i=0}^{2} {2 \choose i} B_i = {2 \choose 0} B_0 + {2 \choose 1} B_1 + {2 \choose 2} B_2\) Thus, \(B_3 = 1 + 2 + 2 = 5\) So the Bell number for n = 3 is: \(B_3 = 5\)

Key Concepts

Combinatorics

Combinatorics is a fascinating field of mathematics that deals with counting, arranging, and analyzing configurations. Imagine trying to figure out how many different ways you can assign tasks to a group of people, or how many possible hands you can be dealt in a card game. These are classic examples where combinatorics steps in. For instance, the Bell numbers, which tell us the number of ways to partition a set, are a powerful tool in combinatorics because they allow us to quantify different scenarios logically.

By using combintorial techniques, you can solve problems involving permutations, combinations, and partitions. Such methods often utilize formulas and numbers that can handle large quantities and complex arrangements, making them indispensable in both pure and applied mathematics.

Binomial Coefficients

Binomial coefficients are a crucial component in combinatorics, and they often appear in the expansion of binomial expressions, hence their name. As part of the theory of combinations, binomial coefficients provide a way to choose subsets from a larger set, a task often referred to as "n choose k."

In the context of Bell numbers, the binomial coefficient \({n-1 \choose i}\) is used to weigh different parts of the recursive formula. It defines the number of ways to choose \(i\) elements from \(n-1\) elements without considering the order of selection. This coefficient can be calculated using the formula:

• \(\frac{(n-1)!}{i!(n-1-i)!}\)

Learning how to compute these coefficients efficiently can help in solving many problems involving combinations and permutations, such as calculating probabilities or finding the number of ways an event can occur.

Recursive Formula

A recursive formula is a formula that defines each term of a sequence using the preceding ones. This method is particularly relevant in mathematical sequences or structures that grow based on previous stages. For functions or numbers defined recursively, the next value in the sequence is based on the combination or transformation of previous values.

You can see a stellar example of this in the recursive formula for Bell numbers:

• \(B_n = \sum_{i=0}^{n-1} {n-1 \choose i} B_i\)

Here, each Bell number \(B_n\) is calculated using the sum of weighted previous Bell numbers, weighted by the binomial coefficients. Understanding how these types of formulas work is essential for successfully analyzing series or mathematical processes that build upon themselves over iterative steps.

Factorial

The factorial function, represented by an exclamation mark \((!\)), is a core concept in mathematics, particularly within combinatorics. It refers to the product of all positive integers up to a specified number \(n\). For example, \(5!\) is equal to \(5 \times 4 \times 3 \times 2 \times 1\), which equals 120.

Factorials are vital in calculating combinations and permutations, where the order of elements is paramount. They provide the foundation for computing binomial coefficients, as seen in their formula:

• \(\frac{(n-1)!}{i!(n-1-i)!}\)

Without factorials, determining the number of ways to arrange or select items from a set would be challenging. As factorials grow rapidly with increasing \(n\), they are particularly important when calculating large or complex combinatorial problems.