Pascal's triangle

The Pascal's triangle, named after Blaise Pascal, a famous french mathematician and philosopher, is shown below with 5 rows.

Pascal's triangle

There is a nice Pascal's triangle calculator on this page that you can play with in order to see the Pascal's triangle for up to 99 rows

 Some Important things to notice    

  • The first row starts with 1.
  • Starting with row #2 and every single row after, the row starts with 1 and ends with 1.
  • The number 2 in the third row is found by adding 1 and 1 in the second row.
  • The two 3s in the fourth row is found by adding 1 and 2 and 2 and 1 in the third row.
  • The number 4 in the fifth row is found by adding 1 and 3 in the fourth row. The number 6 in the fifth row in found by adding 3 and 3 in the fourth row. 

You can indeed keep building more rows by doing just that. It is not that complicated. If we add another row, this is how the Pascal's triangle will look like.

                                                 1

                                           1            1

                                      1           2            1

                               1            3           3             1 

                         1            4           6            4             1

                    1          5           10          10            5            1

                            1

                      1           1

                 1          2           1

            1         3           3           1 

       1         4           6            4          1  

  1        5           10         10          5          1

The Pascal's triangle can also be obtained by pulling out the coefficients of (a + b)n

We show the expansion of 4 binomials when n = 0, 1, 2, and 3.

You can really see the coefficients when n = 1, 2, and 3. When n = 0, it is just the number 1 that we put on top on row #1.

(a + b)0 = 1

(a + b)1 = a + b = 1a + 1b

(a + b)2 = a2 + 2ab + b2 = 1a2 + 2ab + 1b2

a3 + 3a2b + 3ab2 + b3 = 1a3 + 3a2b + 3ab2 + 1b3

Pascal's triangle calculator

This calculator will first show the expansion (a + b)n followed immediately by the triangle.

Enter a positive integer:

 

Recent Articles

  1. 45-45-90 Triangle

    May 01, 23 07:00 AM

    45-45-90 triangle
    What is a 45-45-90 triangle? Definition, proof, area, and easy to follow real-world examples.

    Read More

  2. Theoretical Probability - Definition, Explanation, and Examples

    Apr 24, 23 07:02 AM

    Theoretical probability versus experimental probability
    Learn how to compute the likelihood or probability of an event using the theoretical probability formula.

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes