Pascal's Triangle

Pascal’s triangle is an arrangement of numbers in a triangular pattern. It is an infinite array of numbers arranged in rows, and each row corresponds to the coefficients found in the expansion of a binomial.

It begins with 1, and each entry is obtained by adding the two numbers directly above it from the previous row.

Each entry in Pascal’s triangle can also be found using the binomial coefficient triangle formula: 

kn = n!/k! (n - k)!, where n is the row number and k is the position in the row.

For constructing Pascal’s triangle, follow the steps given below:

Step 1: Begin the table with 1. The first row consists of only 1.

Step 2: To draw each new row, increment the start and end by 1. Each interior number is the sum of the two numbers directly above it. For example, 1 + 1 = 2, 2 + 1 = 3, etc.

Step 3: Repeat the same process to create as many rows as you want, as shown below.

Pascal's Triangle Binomial Expansion

Pascal’s triangle provides the coefficients in the expansion of a binomial expression. For example, (x + y)² expands to x² + 2xy + y². The numbers 1, 2, 1 are taken from the second row of Pascal’s triangle. The same rule works for higher powers, so you can quickly find coefficients without expanding each term manually.  

Pascal’s triangle gives us the number we need for the binomial expression of (x + y)ⁿ. We can calculate (x + y)ⁿ using,

(x + y)ⁿ = a₀xⁿy⁰ + a₁x^n-1y¹ + a₂x^n-2y² + … + a_{n - 1}x¹y^{n - 1} + a_nx⁰yⁿ. 

The coefficients a₀, a₁, a₂, etc., come from row n of Pascal’s triangle. 

In probability, Pascal’s triangle is used to find the number of combinations. It helps us find out how many ways something can happen. 

If we are flipping two coins, the possible outcomes are HH, HT, TH, and TT. Now, let us count how many times we get those chances.

• If there are 0 heads: TT (1 possible way)

1 head: TH, HT (2 possible ways)

2 heads: HH (1 possible way)

Now we get the chances as: 1, 2, 1, which matches the 2nd row of Pascal’s triangle. 

• If we flip 3 coins, the outcomes are:

0 head: TTT (1 way)

1 head: HTT, THT, TTH (3 ways)

2 heads: HHT, HTH, THH (3 ways)

3 heads: HHH (1 way)

So we get 1, 3, 3, 1, which is the third row in Pascal’s triangle.

Pascal’s triangle has various patterns; a few of the patterns are explained below.

• Pattern 1: Sum of each row = 2ⁿ

The sum of the values in the n row is equal to 2ⁿ. Here, ‘n’ represents the row index starting from row 0.

Row 0: 1 = 2⁰ = 1

Row 1: 1 + 1 = 2¹ = 2

Row 2: 1 + 2 + 1 = 2² = 4

Row 3: 1 + 3 + 3 + 1 = 2³ = 8

Row 4: 1 + 4 + 6 + 4 + 1 = 2⁴ = 16

• Pattern 2: Prime number

In Pascal’s triangle, if the second number in a row is a prime, then every number between the two 1s in that row is divisible by that prime. 

Example:

Row 5: 1, 5, 10, 10, 5, 1

Here, 5 is a prime number, and all the numbers between the 1s are divisible by 5. 

• Pattern 3: Diagonals are Fibonacci numbers

When we add the numbers in a diagonal slanting down to the left, we will get a Fibonacci series. 

1

1

1 + 1 = 2

1 + 2 = 3

1 + 3 + 1 = 5

1 + 4 + 3 = 8

1 + 5 + 6 + 1 = 13 and so on.

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,...

Other diagonal patterns: Pascal’s triangle contains several diagonal patterns, such as:

The first diagonal is 1s.

The second diagonal is the counting numbers 1, 2, 3,...

The third diagonal is the triangle numbers 1, 3, 6, 10, 15,...

Pascal’s Triangle is a fascinating number pattern that not only helps in combinatorics but also in algebra, probability, and number theory. Here are some useful tips and tricks for students to easily understand Pascal’s triangle. 

• Recognize the pattern: Each number in Pascal’s Triangle is the sum of the two numbers directly above it. Start by memorizing the first few rows to see how the triangle grows.
   
• Use symmetry: The triangle is symmetric. You only need to calculate half the numbers in a row; the other half mirrors it.
   
• Identify the row as n: The nth row corresponds to the coefficients of (a+b)ⁿ in binomial expansions. This connection simplifies many algebra problems.
   
• Highlight special numbers: Notice that the edges of the triangle are all 1s, and the second diagonal gives natural numbers, the third gives triangular numbers, etc. Recognizing these patterns makes calculations quicker.
   
• Practice combinations: Each number represents a combination rn. Practice calculating these using rn=n! / r!(n−r)! to strengthen understanding.
   
• Spot quick tricks: Sum of the numbers in the nth row equals 2ⁿ. This handy trick helps in checking answers and solving problems efficiently.

Pascal’s triangle is used to solve many problems. Below are some applications of Pascal’s triangle across various fields.

• Algebra: In algebra, Pascal’s triangle is used for expanding expressions like (a + b)ⁿ. It is used in algebra, physics, engineering, and computer science.
   
• Probability and Statistics: Pascal’s triangle is used to calculate combinations (nCr), which are used to find probabilities of predicting chances of drawing cards, flipping coins, etc.
   
• Computer Algorithms: In recursive functions, generating patterns, and dynamic programming, Pascal’s triangle is used in code to solve problems and data structures.
   
• Finance: In finance, it is used in calculating risk and investment scenarios. Pascal’s triangle helps with decision-making trees and outcome modeling.
   
• Predicting patterns in mathematics and nature: Pascal’s Triangle reveals patterns like Fibonacci numbers, powers of 2, and triangular numbers, all of which appear in natural growth and arrangements (like petals, pine cones, or shells).