What's the connection between Pascal's triangle and the binomial coefficients?

Let's review. A binomial is a polynomial expression with two terms, such as:

 x + y x2 + 1 x4 - 3x

Binomial expansion refers to a formula by which one can "expand out" expressions like (x+y)5, where the entire binomial is raised to a power.

Look at the binomial x+y and its powers:

• To find (x+y)2 we multiply (x+y) times (x+y), as shown in the following table.

• To find (x+y)3 we multiply (x+y) times (x+y) times (x+y), or we can multiply the answer to (x+y)2 by (x+y), as shown below.

• To find (x+y)4 we multiply (x+y) times (x+y) times (x+y) times (x+y), or we can multiply the answer to (x+y)3 by (x+y).

 (x + y)2  = (x + y)3  = (x + y)4  = x + y    x + y --------------   x2 + xy        xy + y2 ---------------  x2 + 2xy + y2 x2 + 2xy + y2        x + y --------------------- x3 + 2x2y + xy2       x2y + 2xy2 + y3 ---------------------- x3 + 3x2y + 3xy2 + y3 x3 + 3x2y + 3xy2 + y3                x + y ------------------------- x4 + 3x3y + 3x2y2 +  xy3       x3y + 3x2y2 + 3xy3 + y4 ------------------------------ x4 + 4x3y + 6x2y2  + 4xy3 + y4

#### The results:

(x+y)0 = 1

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

(x+y)2 = 1x2 + 2xy + 1y2

(x+y)3 = 1x3 + 3x2y + 3xy2 + 1y3

(x+y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 +1y4

(x+y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 +1y5

Now use the exponent to indicate a row number and look at the binomial expansion again:

 Row Number Binomial Expansion 0 1 1 1x + 1y 2 1x2 + 2xy + 1y2 3 1x3 + 3x2y + 3xy2 + 1y3 4 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4 5 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5

Remove the variables and look only at the coefficients:

 Row Number Binomial Expansion 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1

You are looking at rows 0 through 5 of Pascal's triangle! The numbers in the rows of Pascal's triangle are the same as the coefficients generated by raising the binomial (x+y) to a power.

We can use the terms in Pascal's triangle to write the expansion of any binomial (x+y)n.

 k = 1 k = 2 k = 3 k = 4 k = 5 (x+y)1: 1 1 (x+y)2: 1 2 1 (x+y)3: 1 3 3 1 (x+y)4: 1 4 6 4 1 ...

To see the expansion of (x+y)4 look at row 4 of Pascal's triangle:

 1 Row 0 1 1 Row 1 1 2 1 Row 2 1 3 3 1 Row 3 1 4 6 4 1 Row 4 1 5 10 10 5 1 Row 5 1 6 15 20 15 6 1 Row 6 1 7 21 35 35 21 7 1 Row 7 1 8 28 56 70 56 28 8 1 Row 8

(x+y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

References

1. Ask Dr. Math Archives: Binomial Expansions and Pascal's Triangle
2. Ask Dr. Math FAQ: Pascal's Triangle
3. Eric's Treasure Trove of Mathematics: Binomial Coefficient