Problem 25 - Discrete Mathematics with Applications

Question

The nth Catalan number \(\mathrm{C}_{n},\) named after the Belgian mathematician, Eugene Charles Catalan ( \(1814-1894 ),\) is defined by $$ \mathrm{C}_{n}=\frac{(2 n) !}{n !(n+1) !}, \quad n \geq 0 $$ where \(n !(n \text { factorial) is defined by } n !=n(n-1) \ldots 3 \cdot 2 \cdot 1 \text { and } 0 !=1 .\) Catalan numbers have many interesting applications in computer science. For example, the number of well-formed sequences of \(n\) pairs of left and right parentheses is given by the \(n\) th Catalan number. Compute the number of legally paired sequences with the given pairs of left and right parentheses. Five

Step-by-Step Solution

Answer

The number of legally paired sequences with 5 pairs of left and right parentheses is 42.

Step 1: Write down the nth Catalan number formula

We have the formula for the nth Catalan number as follows: \[\mathrm{C}_{n} = \frac{(2n)!}{n!(n+1)!}, \quad n \geq 0\]

Step 2: Substitute n with 5

We are given n=5 pairs of parentheses, substitute n with 5 in the formula: \[\mathrm{C}_{5} = \frac{(2*5)!}{5!(5+1)!}\]

Step 3: Calculate Factorials

Calculate the factorial values as follows: 1. (2*5)! = 10! = 3,628,800 2. 5! = 120 3. (5+1)! = 6! = 720

Step 4: Substitute Factorial values into the formula

Use the factorial values in the formula: \[\mathrm{C}_{5} = \frac{3,628,800}{120 \cdot 720}\]

Step 5: Calculate the number of pairings

Now, let's calculate the value of \(\mathrm{C}_{5}\) by performing the division: \[\mathrm{C}_{5} = \frac{3,628,800}{120 \cdot 720} = 42\] So, there are 42 legally paired sequences with 5 pairs of left and right parentheses.

Key Concepts

Factorials in Mathematics

Understanding factorials is essential for solving a wide range of problems in mathematics, particularly in combinatorics and discrete mathematics. A factorial, denoted by an exclamation point, is the product of all positive integers up to a given number. For example, the factorial of 5, written as \(5!\), is calculated by multiplying all positive integers from 1 to 5: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

The concept of factorials becomes especially crucial when dealing with permutations and combinations, where you need to determine the number of ways to arrange or select items. In our exercise, computing the factorial of numbers was a key step in finding the Catalan number for \(n=5\).

The practical application in this scenario helps to count the number of distinct ways to pair parentheses. It's important to grasp how fast factorials grow as the number increases, which underscores their significant impact on the outcome of combinatorial expressions.

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns in sets, usually of finite size. It's known for its applications in probability, statistics, and algorithm design. Within combinatorics, one encounters various concepts, including permutations (arrangements of objects), combinations (selections of objects), and, as seen in our exercise, Catalan numbers.

Catalan numbers are a sequence of natural numbers that appear in various counting problems, often reflecting the number of ways to correctly organize or structure certain objects. The calculation of the \(n\)th Catalan number involves factorials, reinforcing their combinatorial relevance. In the case given, finding the number of ways to legally pair \(n\) sets of parentheses is a typical combinatorial problem. Here, understanding how Catalan numbers are derived and computed is vital to solving related exercises and appreciating their broader use in combinatorics.

Applications of Discrete Mathematics

Discrete mathematics encompasses a collection of topics that are foundational to computer science and information theory. This field includes studies on graphs, networks, algorithms, and - as particularly relevant to our exercise - combinatorial structures like the Catalan numbers.

The applications of discrete mathematics are vast. For instance, in computer science, Catalan numbers turn up in binary tree enumeration, sorting algorithm operations, and parsing expressions in compilers. This implies a deep connection between abstract mathematical concepts and practical computational problems. The exercise of counting the number of legally paired sequences has direct implications in programming languages, where such counting ensures the correctness of nested structures, such as parentheses in mathematical expressions and brackets in code.

Recognizing the significance of discrete mathematics in real-world problem-solving encourages students to appreciate concepts like factorials and Catalan numbers not just as theoretical constructs but as tools with direct practical utility.