Born: June 19, 1623 Clermont Auvergne, France Died: 19 Aug 1662 Paris, France

1. Blaise Pascal Born: June 19, 1623 Clermont Auvergne, France Died: 19 Aug 1662 Paris, France

2. 1 1 1 1 2 1 1 5 10 10 5 1 1 1 3 3 1 6 15 20 15 6 1 4 6 4 1 1 1 7 21 35 35 21 7 1 Pascal’s Triangle

3. 1 1 1 2 1 1 1 3 3 4 6 4 1 1 Creating Pascal’s Triangle 1 (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 • Answer the following questions before going to the next slide: • Where do the numbers come from to form the triangle? • How do you continue the triangle for three more rows?

4. Where do the numbers come from to form the triangle? 1 1 1 2 1 1 1 3 3 4 6 4 1 1 1 10 1 10 5 5 20 1 15 15 6 1 6 21 1 7 21 35 35 7 1 The entries in the triangle are the coefficients of the terms of the binomial expression – (x + y)n 1 (x + y)0 = 1 (x + y)1 = x + y 1 1 1 1 (x + y)2 = x 2 + 2xy + y 2 (x + y)3 = x 3 + 3x2y + 3xy 2 + y3 1 1 1 1 (x + y)4 = x 4 + 4x3y + 6x2y 2 + 4xy3 + y4 2) How do you continue the triangle for three more rows? 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:

5. 1 1 1 1 2 1 1 5 10 10 5 1 1 1 3 3 1 6 15 20 15 6 1 4 6 4 1 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? The answers are all perfect squares. Try to find some other number patterns.

6. 1 1 1 1 2 1 1 5 10 10 5 1 1 1 3 3 1 6 15 20 15 6 1 4 6 4 1 1 1 7 21 35 35 21 7 1 Patterns in Pascal’s Triangle 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? Some people call this the Hockey Stick Pattern. Try this for other parts of diagonals?

7. Formula: nCr = 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) 3C2 = 3! / (2!1!) = (3*2*1) / (2*1*1) = 6/2 = 3

8. 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

9. 1 1 1 1 2 1 1 way for choosing three socks 1 3 3 1 3 ways for choosing one sock 3 ways for choosing two socks Compare using the previous formula method. nCr = 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. • Ignore the 1’s running down the left-hand side of Pascal’s Triangle. • Choose the row where 3 is in the second position because you have 3 socks.

10. 10C5 = = 252 ways to choose five players 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. Or…

11. 10C5 = = 252 Use Pascal’s Triangle to find the number of ways. 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. Pascal’s Triangle and the formula result in the same answer.

12. 8C3 = = 56 ways to choose five players 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. OR …

13. 8C3 = = 56 Use Pascal’s Triangle to find the number of ways. 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. Pascal’s triangle gives the same answer as the formula.

14. 1 1 1 1 2 1 Fun with Pascal’s Triangle Suppose your name is Tom Arrange your name in Pascal’s Triangle form with one letter per line. Find how many different paths you can take to spell your name. T O O M M M 1 1 2 The name arrangement is like Pascal’s Triangle. 1 path ends with the M on the left 2 paths end with the M in the middle The number of letters in your name determines how many rows you use. 1 path ends with the M on the right. There are 4 paths: black, red; black purple; blueorange; blue, green The sum of the numbers in the last row tells you how many paths will spell your name.

15. I’ll try my name. You try your name. Suppose I only want to count the part that is not covered by the triangles? How many ways will there be? 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 70 ways 1 7 21 35 35 21 7 1 How many paths spell Gronberg? 128 ways

16. Fun and Games Use the website to learn how to make a “pinball” machine, as in the picture, based on Pascal’s Triangle. http://www.math.bcit.ca/entertainment/pascaltr/index.shtml

17. The Pinball Game 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. • 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. 1 4 6 4 1 Number of path ending in each bin. The 5th row of Pascal’s Triangle because there are 5 bins.