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Pascal's Triangle

By Dana Valdez

Learn about what Pascal’s Triangle is and how to use it to solve problems with expanding binomials and combinatorial problems. Find the definition, example problems, and practice problems at Thinkster Math.

Why is this concept useful?

Pascal’s Triangle is used to solve problems involving expanding binomials. Using this method provides a quicker way to expand binomials. Pascal’s Triangle can also be used to solve problems involving the Fibonacci Sequence. Another way to use Pascal's Triangle is for combinatorial problems, such as probability.

Where does this concept fit into the curriculum?

High School Algebra

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                   

(x+y)0(x + y)^0

1   1                

(x+y)1(x + y)^1

1   2  1              

(x+y)2 (x + y)^2

1  3   3  1            

(x+y)3(x + y)^3

1  4   6   4  1         

(x+y)4(x + y)^4

1   5  10 10  5  1      

(x+y)5(x + y)^5

1   6  15 20 15  6  1   

(x+y)6(x + y)^6

The top row is considered the 0th row, and represents an exponent of 0.

For example:

(x+y)0=1(x + y)^0 = 1

The second row is considered the 1st row and represents an exponent of 1.

For example:

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

The third row is considered the 2nd row and represents an exponent of 2.

For example:

(x+y)2(x + y)^2
=
x2+2xy+y2x^2 + 2xy + y^2

The fourth row is considered the 3rd row and represents an exponent of 3.

For example:

(x+y)3(x + y)^3
=
x3+3x2yx^3 + 3x^2y
+3xy2+y3+ 3xy^2 + y^3

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

(x+5)3(x + 5)^3

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)

(x+5)3(x + 5)^3
=
1x3(5)01x^3(5)^0
+3x2(5)1+ 3x^2(5)^1
+3x1(5)2+3x^1(5)^2
 
+1x0(5)3+ 1x^0(5)^3

Next, simplify:

x3+15x2x^3 + 15x^2
+ 75x + 125

Question

Expand

(x+1)3(x + 1)^3

Solution

Use the row in Pascal’s Triangle that represents the power of 3 (the row which has 3 as its second number). The numbers in this row will be the coefficients (1  3  3  1)

(x+1)3(x + 1)^3
=
1x3+3x21x^3 + 3x^2
+3x+1+ 3x + 1

Question

Use Pascal’s Triangle to find the 9th term in Fibanacci’s Sequence.

Continue the pattern in Pascal’s Triangle, the digits in the 9th diagonal are:

Solution

1, 10, 15, 7, 1

Next, find the sum of the digits 1 + 10 + 15 + 7 + 1 = 34, so the 9th digit in Fibonacci’s Sequence is 34.

Question

If you are making a smoothie and have 5 different fruits, how many combinations of 3 fruits could you use?

To use Pascal’s Triangle, simply go to the row that represents the 5th power and find the 3rd number.

Solution

The 5th line is: 1 5 10 10 5 1 and the third number is 10. 10 is the correct answer.

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Practice Math Problems

1. Now use Pascal’s Triangle to expand the following binomials:

(x+4)4(x + 4)^4

2. Now use Pascal’s Triangle to expand the following binomials

(x+y)3(x + y)^3

3. Use Pascal’s Triangle to find the 10th term in the Fibonacci Sequence

4. If you have 10 beads of different colors, how many combinations of 4 colors could be made?

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