Born: June 19, 1623 Clermont Auvergne, France

Died: 19 Aug 1662 Paris, France

Blaise Pascal

Pascal’s Triangle

1

1 1

121

33 11

4641 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Creating Pascal’s Triangle

Answer the following questions before going to the next slide:

1) Where do the numbers come from to form the triangle?

2) How do you continue the triangle for three more rows?

(x + y)0 = 1

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

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

(x + y)3 = 1x 3 + 3x2y + 3xy 2 + 1y3

(x + y)4 = 1x 4 + 4x3y + 6x2y 2 + 4xy3 + 1y4

1

1 1

121

33 11

4641 1

1) Where do the numbers come from to form the triangle?

2) How do you continue the triangle for three more rows?

The entries in the triangle are the coefficients of the terms of the binomial expression – (x + y)n

(x + y)0 = 1

(x + y)1 = x + y

(x + y)2 = x 2 + 2xy + y 2

(x + y)3 = x 3 + 3x2y + 3xy 2 + y3

(x + y)4 = x 4 + 4x3y + 6x2y 2 + 4xy3 + y4

1

1 1

121

33 11

4641 1

In the triangle the first and last terms are always 1. The terms in between are the sum of two adjacent terms. (watch above)..

The next three rows would be:105 51 10 1

1 6 15 20 15 6 1

1 7 3521 35 21 7 1

1 1

1 1

1 1

1 1

1

1 1

121

33 11

4641 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Patterns in Pascal’s Triangle

Look at the row in the blue ellipse.

Add the numbers two at a time.

Example: 1 + 3 = 4

3 + 6 = 9

6 +10 =16

What kind of answers do you get?

Try to find some other number patterns.

The answers are all perfect squares.

1

1 1

121

33 11

4641 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Look at the part of a diagonal in the rectangle.

Add the three numbers together.

1 + 3 + 6 + 10 = 20

Where do you find the answer in Pascal’s Triangle?

Patterns in Pascal’s Triangle

Try this for other parts of diagonals?

Some people call this the Hockey Stick Pattern.

Combinations Using Formulas

1. Suppose you have three socks and want to figure out how many different ways you can choose two of them to wear. You don’t care which feet you put them on, it only matters which two socks you pick. This problem amounts to the question “how may different ways can you choose two objects from a set of three objects?”

Suppose you have S1, S2, S3 (sock 1, sock 2, sock 3)

You could choose: S1,S2 S1,S3 S2,S3 (3 ways to choose 2 socks)

Formula: nCr = r!r)!(n

n!

3C2 = 3! / (2!1!) = (3*2*1) / (2*1*1) = 6/2 = 3

Combinations Using Formulas (continued)

Suppose you wanted to choose one sock. S1 or S2 or S3 (3 ways)

3C1 = 3! / 1!2! = 3 ways

Suppose you wanted to choose all three socks {S1, S2, S3} (1 way)

3C3 = 3! / 3!0! = 1 way

Combinations and Pascal’s Triangle

Now use Pascal’s Triangle to determine how many ways you can choose 1 sock, 2 socks, or 3 socks.

11 1

12133 11

1) Ignore the 1’s running down the left-hand side of Pascal’s Triangle.

2) Choose the row where 3 is in the second position because you have 3 socks.

3 ways for choosing one sock

3 ways for choosing two socks

1 way for choosing three socks

Compare using the previous formula method.nCr = r!r)!(n

n!

A basketball coach is criticized in the newspaper for not trying out every combination of players. If the team roster has 10 players, how many five-player combinations are possible?

Use can use the combination formula to find how many combinations of five players are possible.

5!5!

10!10C5 = = 252 ways to choose five players

Or…

You choose the row with 10 after the 1 because there are 10 players from which to choose.

The row where 10 is the first number after the 1 is:

1 10 45 120 210 252 210 120 45 10 1

This row tells you there are:

10 ways to choose one player. 210 ways to choose four players.

45 ways to choose two players. 252 ways to choose five players.

120 ways to choose three players.

Use Pascal’s Triangle to find the number of ways.

5!5!

10!10C5 = = 252 Pascal’s Triangle and the formula

result in the same answer.

There are 8 people in an office. 3 are to be chosen for the grievance committee. In how many ways can 3 people be chosen out of the eight in the office?

You can use the combination formula to find how many combinations of three people are possible.

5!3!

8!8C3 = = 56 ways to choose five players

OR …

You choose the first row that has an 8 after the one since there are 8 people to choose from.

The row where 8 (you have 8 people) is the first number after the 1:

1 8 28 56 70 56 28 8 1

This row of numbers tells you:

8 ways to choose one person.

28 ways to choose two people.

56 ways to choose three people.

Use Pascal’s Triangle to find the number of ways.

5!3!

8!8C3 = = 56

Pascal’s triangle gives the same answer as the formula.

Fun with Pascal’s Triangle

Find how many different paths you can take to spell your name.Arrange your name in Pascal’s Triangle form with one letter per line.

Suppose your name is Tom

T

O O

M M M

There are 4 paths:

black, red; black purple;

blue orange; blue, green

1 path ends with the M on the left

1

1 1

1211 2 1

2 paths end with the M in the middle

1 path ends with the M on the right.

The name arrangement is like Pascal’s Triangle.

The number of letters in your name determines how many rows you use.

The sum of the numbers in the last row tells you how many paths will spell your name.

I’ll try my name. You try your name.

G

R R

O O O

N N N N

B B B B B

E E E E E E

R R R R R R R

G G G G G G G G

1 7 21 35 35 21 7 1

How many paths spell Gronberg?

128 ways

Suppose I only want to count the part that is not covered by the triangles?

How many ways will there be?

70 ways

http://www.math.bcit.ca/entertainment/pascaltr/index.shtml

Use the website to learn how to make a “pinball” machine, as in the picture, based on Pascal’s Triangle.

Fun and Games

The Pinball Game

1 4 6 4 1

The gray lines indicate the paths. There are 5 bins where the marble can finish.

What is the probability that the ball will end up in any given bin?

There are 16 paths (sum of the number of paths to each bin) and the ball is equally likely to follow any of the paths.

Number of path ending in each bin. The 5th row of Pascal’s Triangle because there are 5 bins.

•Probability = 1/16 to end in the 1st (leftmost) bin.•Probability = 4/16 to end in the 2nd bin.•Probability = 6/16 to end in 3rd bin.•Probability = 4/16 to end in 4th bin.•Probability = 1/16 to end in the 5th bin.