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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:

 
(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.

                    Coefficient in the (k)th term:
 

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

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