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Wize High School Grade 11 Math Textbook > Sequences & Series

Pascal's Triangle and the Binomial Expansion

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Pascal's Triangle

Pascal's Triangle is made by starting with a single 1 in the top row, then two 1s in the next row.
For each of the next rows, each number is the sum of the two numbers directly above it.



Wize Tip
  • Every row is symmetric
  • The rows are numbered starting with row 0
  • You can tell which row you are looking at by the second entry: the second entry of row nn is always nn, e.g. the second entry of row 3 is 3.

Binomial Expansion

One application of the Pascal's Triangle is for binomial expansions.

A binomial expansion is the result of expanding expressions of the form (x+y)n\left(x+y\right)^n.

(x+y)n=a1xny0+a2xn−1y1+a3xn−2y2+...+an+1x0yn\Large{\boxed{(x+y)^n=\colorFive{a_1}\colorThree{x^n}\colorTwo{y^0}+\colorFive{a_2}\colorThree{x^{n-1}}\colorTwo{y^1}+\colorFive{a_3}\colorThree{x^{n-2}}\colorTwo{y^2}+...+\colorFive{a_{n+1}}\colorThree{x^0}\colorTwo{y^n}}}

The coefficients a1, a2, ...,an\colorTwo{a_1,~a_2,~...,a_n} are the numbers in the nth row of the Pascal's Triangle. (remember that the triangle starts with row 0)


Example

Expand (x+y)n(x+y)^n for n=1,2,3n=1,2,3. Notice how the result relates to Pascal's triangle.

n=1\colorOne{\bm{n=1}} case:


(x+y)=1x1y0+1x0y1=x+y\begin{array}{ccccc} (x+y) &= &\colorFive1\colorThree{x^1}\colorTwo{y^0} &+ &\colorFive1\colorThree{x^0}\colorTwo{y^1}\\ &= &x &+ &y \end{array}

n=2\colorOne{\bm{n=2}} case:


(x+y)2=1x2y0+2x1y1+1x0y2=x2+2xy+y2\begin{array}{ccccccc} (x+y)^2 &= &\colorFive{1}\colorThree{x^2}\colorTwo{y^0} &+ &\colorFive{2}\colorThree{x^1}\colorTwo{y^1} &+ &\colorFive{1}\colorThree{x^0}\colorTwo{y^2}\\ &= &x^2 &+ &2xy &+ &y^2 \end{array}

n=3\colorOne{\bm{n=3}} case:


(x+y)3=1x3y0+3x2y1+3x1y2+1x0y3=x3+3x2y+3xy2+x3\begin{array}{ccccccc} (x+y)^3 &= &\colorFive{1}\colorThree{x^3}\colorTwo{y^0} &+ &\colorFive{3}\colorThree{x^2}\colorTwo{y^1} &+ &\colorFive{3}\colorThree{x^1}\colorTwo{y^2} &+ &\colorFive{1}\colorThree{x^0}\colorTwo{y^3}\\ &= &x^3 &+ &3x^2y &+ &3xy^2 &+ &x^3 \end{array}
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Example: Pascal's Triangle and the Binomial Expansion

Expand and simplify (x+2y)5(x+2y)^5 using Pascal's triangle.

Step 1: Write out the n+1 row of the Pascal's triangle

We need to write out the 5th row of the Pascal's triangle:
(*remember that the triangle starts with row 0)

Step 2: Write out the binomial expansion of the form (x+y)n=a1xny0+a2xn−1y1+a3xn−2y2+...+an+1x0yn\boxed{(x+y)^n=\colorFive{a_1}\colorThree{x^n}\colorTwo{y^0}+\colorFive{a_2}\colorThree{x^{n-1}}\colorTwo{y^1}+\colorFive{a_3}\colorThree{x^{n-2}}\colorTwo{y^2}+...+\colorFive{a_{n+1}}\colorThree{x^0}\colorTwo{y^n}}

Notice that the first term inside the brackets is xx and the second term is 2y2y:

➡ Fill in the coefficients a1,a2,...,an+1\colorFive{a_1},\colorFive{a_2},...,\colorFive{a_{n+1}} using Pascal's triangle:
(x+2y)5=1...+5...+10...+10...+5...+1...\begin{array}{ccccccc} (x+2y)^5 &= &\colorFive{1}... &+ &\colorFive{5}... &+ &\colorFive{10}... &+ &\colorFive{10}... &+ &\colorFive{5}... &+ &\colorFive{1}... \end{array}


➡ Fill in the xn,xn−1,...,x0\colorThree{x^n},\colorThree{x^{n-1}},...,\colorThree{x^0} terms:
(x+2y)5=1x5...+5x4...+10x3...+10x2...+5x1...+1x0...\begin{array}{ccccccc} (x+2y)^5 &= &\colorFive{1}\colorThree{x^5}... &+ &\colorFive{5}\colorThree{x^4}... &+ &\colorFive{10}\colorThree{x^3}... &+ &\colorFive{10}\colorThree{x^2}... &+ &\colorFive{5}\colorThree{x^1}... &+ &\colorFive{1}\colorThree{x^0}... \end{array}


➡ Fill in the (2y)0,(2y)1,...,(2y)n\colorTwo{(2y)^0},\colorTwo{(2y)^1},...,\colorTwo{(2y)^{n}} terms:
(x+2y)5=1x5(2y)0+5x4(2y)1+10x3(2y)2+10x2(2y)3+5x1(2y)4+1x0(2y)5\begin{array}{ccccccc} (x+2y)^5 &= &\colorFive{1}\colorThree{x^5}\colorTwo{(2y)^0} &+ &\colorFive{5}\colorThree{x^4}\colorTwo{(2y)^1} &+ &\colorFive{10}\colorThree{x^3}\colorTwo{(2y)^2} &+ &\colorFive{10}\colorThree{x^2}\colorTwo{(2y)^3} &+ &\colorFive{5}\colorThree{x^1}\colorTwo{(2y)^4} &+ &\colorFive{1}\colorThree{x^0}\colorTwo{(2y)^5} \end{array}


Step 3: Simplify

(x+2y)5=x5+10x4y+40x3y2+80x2y3+80xy4+32y5\boxed{\begin{array}{ccccccc} (x+2y)^5 &= &x^5 &+ &10x^4y &+ &40x^3y^2 &+ &80x^2y^3 &+ &80xy^4 &+ &32y^5 \end{array}}

Practice: Pascal's Triangle and the Binomial Expansion

Match each of the following powers of binomials with the appropriate row of Pascal's triangle.
A.
146411 \quad 4 \quad 6 \quad 4 \quad 1
B.
13311 \quad 3 \quad 3 \quad 1
C.
1211 \quad 2 \quad 1
D.
151010511 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1
(3x−5)4(3x-5)^4
(−2x+4y)3(-2x+4y)^3
(5x+3y)2(5x+3y)^2
(2−2y)5(2-2y)^5

Practice: Pascal's Triangle and the Binomial Expansion

a) Expand and simplify (3x−1)4(3x-1)^4

b) Expand and simplify the first three terms of (5a−4b)6(5a-4b)^6

Practice: Pascal's Triangle and the Binomial Expansion

How many possible ways are there to move the red checker down to the highlighted square?


Note: a checker can only move diagonally by a single space (no jumping).