2. Whiteboardmaths.com © 2004 All rights reserved 5 7 2 1

3. Stepping Stone Game Pascal’s Triangle: The Stepping Stone Game How many different routes are there from the Start stone to the Finish stone? Rules: You can only walk East or South from any stone. We will start by looking at 5 possible routes (be careful how you walk)

4. How many routes are there to: 1 1 1 1 1 1 1 1 2 1 Pascal’s Triangle: The Stepping Stone Game

5. How many routes are there to: 1 1 1 1 1 1 1 1 2 3 1 Pascal’s Triangle: The Stepping Stone Game

6. How many routes are there to: 1 1 1 1 1 1 1 1 2 3 3 1 Pascal’s Triangle: The Stepping Stone Game

7. How many routes are there to: 1 1 1 1 1 1 1 1 2 3 3 6 Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualise them? 1 Pascal’s Triangle: The Stepping Stone Game

8. How many routes are there to: 1 1 1 1 1 1 1 1 2 3 3 6 Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualise them? 3 routes to this stone What do you have to do to get the number of routes to any stone? Why must there be 6 routes to here? 1 Pascal’s Triangle: The Stepping Stone Game

9. How many routes are there to: 1 1 1 1 1 1 1 1 2 3 3 6 Can you see all 6 of the routes? How could you have calculated the 6 routes without the need to draw or visualising them? What do you have to do to get the number of routes to any stone? 4 5 Calculate the total number of routes to the finish stone. 10 15 4 5 10 15 20 35 70 1 Pascal’s Triangle: The Stepping Stone Game

10. 1 1 1 1 1 1 1 1 2 3 3 6 4 5 10 15 4 5 10 15 20 35 70 1 Do you notice anything about the numbers produced by the routes through to the finish stone? The numbers are symmetrical about the diagonal line. Counting numbers Triangular numbers Tetrahedral numbers 1 514 30 Square base Pyramid numbers Pascal’s Triangle: The Stepping Stone Game

11. Pascal’s Triangle 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 9 3684 126 84 36 9 10 45 210 252 210 120 45 120 10 11 55 330 462 330 165 55 1112 66 495 792924 792 495 220 66 12 1716 13 78 286 715 1287 1716 1287 715 286 78 13 Pascal’s Triangle 4 8 16 32 64 128 256 512 1024 2048 4096 8192 2 1=2 0 =2 1 =2 2 =2 3 =2 4 =2 5 =2 6 =2 7 =2 8 =2 9 =2 10 =2 11 =2 12 =2 13 2. Find the sum of each row. 1. Complete the rest of the triangle. 3. Write the sum as a power of 2. R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 Blaisé Pascal (1623-1662) Counting/Natural Numbers Triangular Numbers Tetrahedral Numbers Pyramid Numbers (square base)

12. Fibonacci Sequence 1716 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 9 3684 126 84 36 9 10 45 210 252 210 120 45 120 10 11 55 330 462 330 165 55 11 12 66 495 792924 792 495 220 66 12 13 78 286 715 1287 1716 1287 715 286 78 13 Add the numbers shown along each of the shallow diagonals to find another well known sequence of numbers. 1 23813 21 5589 15 34 144 233 377 The sequence first appears as a recreational maths problem about the growth in population of rabbits in book 3 of his famous work, Liber – abaci (the book of the calculator ). Fibonacci travelled extensively throughout the Middle East and elsewhere. He strongly recommended that Europeans adopt the Indo-Arabic system of numerals including the use of a symbol for zero “zephirum” The Fibonacci Sequence Leonardo of Pisa 1180 - 1250

13. Activity: There are 5 books on a shelf. Complete the table to find the number of ways that you can choose 0, 1, 2, 3, 4 and 5 books. ChoosePossibilitiesN o of Ways 0 books - 1 1 book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 4 books ABCD, ABCE ABDE ACDE BCDE 5 books ABCDE You will need to think systematically! A B C D E Books 5

14. ChoosePossibilitiesN o of Ways 0 books - 1 1 book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE DE 10 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 10 4 books ABCD, ABCE ABDE ACDE BCDE 5 5 books ABCDE 1 Activity: There are 5 books on a shelf. Complete the table to find the number of ways that you can choose 0, 1, 2, 3, 4 and 5 books. You will need to think systematically! A B C D E Relate these numbers to entries in Pascal’s triangle.

15. A B C D E R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 012345 1510 51 5 Books The entries in row 5 give the number of combinations of choosing 0,1,2,3,4 and 5 books respectively. Choose 0Choose 1Choose 2 Choose 3 Choose 4 Choose 5 So if you wanted to know the probability of choosing one particular combination of 2 books at random, then the probability is 1/10

16. Activity: There are 6 balls in a box as shown below. Complete the table to find the number of ways that you can choose 0,1,2, 3,4,5 and 6 balls from the box. ChoosePossibilities N o of Ways 0 balls - 1 1 ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 3 balls 123, 124, 125, 126 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456 4 balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 5 balls 12345, 12346, 12356, 12456, 13456, 23456 6 balls 123456 You will need to think even more systematically!

17. Activity: There are 6 balls in a box as shown below. Complete the table to find the number of ways that you can choose 0,1,2, 3,4,5 and 6 balls from the box. ChoosePossibilities N o of Ways 0 balls - 1 1 ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 15 3 balls 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456 20 4 balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456 15 5 balls 12345, 12346, 12356, 12456, 13456, 23456 6 6 balls 1234561 Relate these numbers to entries in Pascal’s triangle. You will need to think even more systematically!

18. R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 0123456 1615201561 6 Balls The entries in row 6 give the number of combinations of choosing 0,1,2,3,4,5 and 6 balls respectively. Choose 0Choose 1Choose 2 Choose 3 Choose 4 Choose 5 Choose 6 So if you wanted to know the probability of choosing one particular combination of 3 balls at random, then the probability is 1/20

19. Balls 12 R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 12 11 10 9 8 7 6 5 4 3 2 1 12 Balls Use Pascal’s triangle to determine how many ways there are of choosing: (a)2 balls (b) 5 balls (c) 9 balls Choose 0 66 792 220 The probability of choosing one particular combination of 5 balls is 1/792 Find the combination key on a scientific calculator and evaluate. nCrnCr 12 C 2 12 C 5 12 C 9 12 choose 2 12 choose 5 12 choose 9 n choose r

20. Books 8 R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 Use Pascal’s triangle to determine how many ways there are of choosing: (a)3 books (b) 4 books (c) 6 books Choose 0 56 70 28 8 Books nCrnCr 8C48C4 8C68C6 8C38C3 Now do it on a calculator. The probability of choosing one particular combination of 4 books is 1/70

21. 5-a-side R0R0 R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 R7R7 R8R8 R9R9 R 10 R 11 R 12 R 13 252 nCrnCr 10 C 5 In how many ways can a 5-a-side team be chosen from a squad of 10 players? Choose 0 The probability of choosing one particular combination of 5 players is 1/252

22. Remember: The top row is Row 0 Use Pascal’s triangle to determine the number of combinations for each of the following selections.

23. Mix 8 7 6 5 4 9 3 2 1 A B C D Choose 3 books Choose 5 players Choose 7 cards Choose 4 balls

24. Lottery 8 7 6 5 4 9 3 2 1 27 26 25 24 23 28 22 21 20 29 17 16 15 14 13 18 12 11 10 19 37 36 35 34 33 38 32 31 30 39 47 46 45 44 43 48 42 41 40 49 National Lottery Jackpot? 49 balls choose 6 ?

25. 8 7 6 5 4 9 3 2 1 27 26 25 24 23 28 22 21 20 29 17 16 15 14 13 18 12 11 10 19 37 36 35 34 33 38 32 31 30 39 47 46 45 44 43 48 42 41 40 49 National Lottery Jackpot? 49 balls choose 6 Choose 6Row 49 13 983 816 There are 13 983 816 ways of choosing 6 balls from a set of 49. So buying a single ticket means that the probability of a win is 1/13 983 816 49 C 6 Row 0

26. Pascal’s Triangle on a Spreadsheet ABCDEFG… 1 2 3 4 5 6 7 8 1. Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. Spreadsheet

27. Pascal’s Triangle on a Spreadsheet ABCDEFG… 11111111 21 31 41 51 61 71 81 1. Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. 2. In cell B2 enter the formula =B1+A2 then fill right.

28. Pascal’s Triangle on a Spreadsheet ABCDEFG… 11111111 21 234567 31 41 51 61 71 81 1. Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds. 2. In cell B2 enter the formula =B1+A2 then fill right. 3. Fill down as far as row 50. Remember that row 1 on the spreadsheet corresponds to row 0 in Pascal’s triangle.

29. Pascal’s Triangle on a Spreadsheet ABCDEFG… 11111111 21 234567 31 3610152128 41 41020355684 51 5153570126210 61 62156126252462 71 72884210462924 81 836120330792 1716 1. Enter 1’s along rows and down columns. Go down to row 50 for jackpot odds 2. In cell B2 enter the formula =B1+A2 then fill right. 3. Fill down as far as row 50. Remember that row 1 on the spreadsheet corresponds to row 0 in Pascal’s triangle. 4. Use a step-up procedure to see each row more clearly. Row 3 Row 6

30. Pascal’s Triangle on a Spreadsheet 421 431 441 451 461 471 481 491 501 13 983 816 Row 49 6 th entry

31. Historical Note Blaisé Pascal (1623-1662) Pascal was a French mathematician whose contemporaries and fellow countrymen included Fermat, Descartes and Mersenne. Among his many achievements was the construction of a mechanical calculating machine to help his father with his business. It was able to add and subtract only, but it was a milestone on the road to the age of computers. He corresponded with Fermat on problems that led to the new branch of mathematics called Probability Theory. The two problems that they examined concerned outcomes when throwing dice and how to divide the stake fairly amongst a group of players if a game was interrupted. These investigations led Pascal to construct tables of probabilities that eventually led to the triangle of probabilities that bears his name. Pierre de Fermat (1601 – 1675)

32. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 The Binomial Expansion Pascal used his triangle to find the coefficients in the expansion of (a + b) n The coefficients of each term correspond to the entries in the n th row. (a + b) 2 = a 2 +2ab +b 2 (a + b) 3 = a 3 +3a 2 b +3ab 2 + b 3 (a + b) 4 = a 4 + 4a 3 b +6a 2 b 2 + 4ab 3 + b 4 (a + b) 5 = a 5 + 5a 4 b +10a 3 b 2 + 10a 2 b 3 + 5ab 4 +b 5 (a + b) 8 = a 8 + a 7 b + a 6 b 2 + a 5 b 3 + a 4 b 4 + a 3 b 5 + a 2 b 6 + ab 7 + b 8 (a + b) 8 = a 8 + 8a 7 b + 28a 6 b 2 + 56a 5 b 3 + 70a 4 b 4 + 56a 3 b 5 + 28a 2 b 6 + 8ab 7 + b 8 Binomial Expansion

33. Worksheet 1 Pascal’s Triangle: The Stepping Stone Game

34. Worksheet 2

35. Worksheet 3 1716 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 33 464 5510 6156 20 7 721 35 88 2856285670 9 3684 126 84 36 9 10 45 210 252 210 120 45 120 10 11 55 330 462 330 165 55 11 12 66 495 792924 792 495 220 66 12 13 78 286 715 1287 1716 1287 715 286 78 13

36. Worksheet 4 ChoosePossibilitiesN o of Ways 0 books - 1 1 book A,B,C,D,E 5 2 books AB, AC,AD, AE BC, BD, BE CD, CE 3 books ABC, ABD, ABE ACD, ACE ADE BCD, BCE BDE, CDE 4 books ABCD, ABCE ABDE ACDE BCDE 5 books ABCDE

37. Worksheet 5 ChoosePossibilities N o of Ways 0 balls - 1 1 ball 1, 2, 3, 4, 5, 66 2 balls 12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, 56 3 balls 123, 124, 125, 126 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456 4 balls 1234, 1235, 1236, 1245, 1246, 1256, 1345, 1346, 1356, 1456, 2345, 2346, 2356, 2456, 3456, 5 balls 12345, 12346, 12356, 12456, 13456, 23456 6 balls 123456