Pascal’s Triangle and its Dedicated Properties

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Blaise Pascal is the mathematician behind the discovery and identification of properties of the arithmetical triangle. Known for his contributions to mathematics, Blaise Pascal gave the world the application of pascals triangle. It refers to a number pattern that is arranged in a triangular format. A single number is placed on the top followed by subsequent numbers arranged in a triangle format. The numbers placed are arrived at with the sum of two numbers above them.

1. Creation of Pascal’s triangle

A Pascal’s triangle can be constructed by following the following process;

  • Start with the zero rows, write the number 1.
  • Next step is the place numbers in the first row. The number placed above is added to the number on the right and left. In case there is no number on the left or right, use the zero number and continue with the addition process. The numbers are arranged in a triangular format where the numbers are placed in the shape of a triangle.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

The above-given figure is a simple example of Pascal’s triangle. It starts with a row 0 and ends at the 5th row. As we move down, the numbers are added and arranged in a triangular format.

2. Use of Pascal’s triangle

Pascal’s triangle is used for calculating the probability of occurrence of various situations. For example, consider the situation of selecting a red ball and blue ball from a bag containing balls of the same color each. The probability of selecting a blue ball is 1 and the red ball is also 1 in case of a single chance. In the case of selection two times, there is a possibility of selecting two red balls, two blue balls, one blue, and red ball, and one red and one blue ball respectively.

The use of Pascal’s triangle for probability can be illustrated as follows:

  1. Probability of selecting blue and red balls in case of a single chance of selection = 1, 1

(B, R)

  1. Probability of selecting blue and red balls in case of two chances of selection = 1, 2, 1

(RR, RB, BR, and BB).

  1. Probability of selecting blue and red balls in case of three chances of selection = 1, 3, 3, 1

(RRR, RRB, RBR, BRR, BBB, BBR, BRB, RBB).

These correspond to the structure as created through Pascal’s triangle.

 

3. Properties of Pascal’s triangle:

There are various dedicated properties of Pascal’s triangle. These are listed below:

  • The number in Pascal’s triangle is the sum of the numbers right above it.
  • The outer diagonal of Pascal’s triangle consists of 1.
  • The sum of numbers in a row of Pascal’s triangle can be written in the form of 2^n where n is the number of rows.
  • Pascal’s triangle is symmetric.
  • Every row of Pascal’s triangle can be expressed in the format of 11^n where n is the number of rows.
  • The first diagonal of Pascal’s triangle shows the counting numbers.

Pascal’s triangle can also be explained as the Binomial expansion. The coefficients of a binomial expansion (x + y)^n serve as the digits that make up rows of Pascal’s triangle. For example, the 2nd row of Pascal’s triangle through Binomial Expression can be explained as follows:

(x + y)^2= 2C0x^2 + 2C1 xy + 2C2 y^2

= (1)x^2 + (2)xy + (1) y^2

The coefficients of the binomial expression form the numbers of the 2nd row of Pascal’s triangle.

Pascal’s triangles, their properties, expressions, and overall concept can be clearly understood. The company makes available an online website from where all the necessary information about Pascal’s triangle can be obtained. Students can get access to all the related information and resources for understanding Pascal’s triangle through its official website. Pascal’s triangle and its dedicated patterns can be understood through practical examples made available on the official website of Cuemath.

 

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