What is Pascal’s Triangle? Pascal’s Triangle is named after a French mathematician, Blaise Pascal. It is a triangle made of patterns, where each number is the sum of the numbers directly above it.
How to Use Pascal’s Triangle: You can use Pascal’s Triangle to expand binomials. Each row of the triangle represents the coefficients used to expand binomials. Here are the first 7 rows of Pascal’s Triangle with the binomial each row represents:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
The top row is considered the 0th row, and represents an exponent of 0.
For example:
The second row is considered the 1st row and represents an exponent of 1.
For example: = x + y
The third row is considered the 2nd row and represents an exponent of 2.
For example: =
The fourth row is considered the 3rd row and represents an exponent of 3.
For example: =
Notice how the coefficients match the corresponding row in Pascal’s Triangle.
Pascal’s Triangle directly relates to the Fibonacci Sequence. The Fibonacci Sequence is a series of numbers where each number is the sum of the two numbers before it. For example, the first ten numbers in the Fibonacci Sequence are: 1, 1, 2, 3, 5, 8,13, 21, 34. To find the 11th term in the sequence, find the sum of 21 and 34, and so on. This directly relates to Pascal's Triangle. The sum of each diagonal in Pascal’s Triangle is the same as the Fibonacci Sequence. Therefore, the nth term in the Fibonacci Sequence is equivalent to the sum of the digits in the nth diagonal of Pascal’s Triangle.
Pascal’s Triangle can also be used to solve problems of probability.
Sample Math Problems
Question
Expand
Solution
Use the row in Pascal’s Triangle that represents the power of 3 (the row which has 3 has its second number). The numbers in this row will be the coefficients (1 3 3 1)
