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3. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely! Before we start consider evaluating the following, involving factorial notation.

4. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. Work out the following without using a calculator. 90 380 2352

5. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. We need to find the total number of ways of choosing 6 balls from a set of 49. Start by considering the number of ways of choosing 1, 2, 3, …… balls. 1 ball 2 balls 3 balls 49 ? ?

6. 1 2 3 4 5 6 7 8 9 Choose 1 (49 ways) 2 Choose 2 (48 ways) 1 3 1 4 1 5 1 6 1 49 1 2 3 2 4 2 5 2 6 2 1 3 2 3 4 3 5 3 6 3 1 4 2 4 3 4 5 4 6 4 1 5 2 5 3 5 4 5 6 5 1 6 2 6 3 6 4 6 5 6 1 23 45 48 49 For each of the 49 ways of choosing the first ball there are 48 ways of choosing the second ball. This gives a total of 49 x 48 ways of choosing 2 balls. Similarly for each of the 49 x 48 ways of choosing 2 balls there are 47 ways of choosing the third ball. So there are 49 x 48 x 47 ways of choosing 3 balls.

7. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. We need to find the total number of ways of choosing 6 balls from a set of 49. Start by considering the number of ways of choosing 1, 2, 3…… balls. 1 ball 2 balls 3 balls 49 49 x 48 49 x 48 x 47 Write the number of ways of choosing 6 balls as

8. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 1 ball 2 balls 49 49 x 48 3 balls49 x 48 x 47 4 balls49 x 48 x 47 x 465 balls49 x 48 x 47 x 46 x 456 balls49 x 48 x 47 x 46 x 45 x 44

9. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 This is a very large number and is more than the number of people on the planet! Nobody would buy a ticket if these were the odds of wining. Can you see what needs to be done to reduce this number and arrive at the true number of possible outcomes?

10. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 CLUE: Consider the order of the balls The above calculation assumes that you have to select the balls in the correct order. Do you have to?

11. 1 2 3 4 5 6 7 8 9 49 Choose 1 (49 ways) 2 Choose 2 (48 ways) 1 3 1 4 1 5 1 6 1 49 1 2 3 2 4 2 5 2 6 2 1 3 2 3 4 3 5 3 6 3 1 4 2 4 3 4 5 4 6 4 1 5 2 5 3 5 4 5 6 5 1 6 2 6 3 6 4 6 5 6 1 23 45 48 49 Since the order of the balls does not matter in the lottery, for a 2 ball game we would have to divide the 48 x 49 outcomes by 2! (the number of ways of arranging 2 objects). Likewise 3! for a 3 ball game … 6! for a 6 ball game. 2 balls49 x 48

12. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 How many different arrangements of 6 balls are there? 6! = 720 We need to make the above number 720 times smaller

13. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 Write the total number of ways (were the order does not matter) as:

14. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 This number is the total number of ways in which 6 balls can be chosen from a set of 49. So you have a 1 in 13 983 816 chance of winning the lottery if you purchase 1 ticket.

15. National Lottery: Jackpot Odds 6 15 26 28 35 49 It could be you!But its highly unlikely. 6 balls49 x 48 x 47 x 46 x 45 x 44 Combinations A combination is an un-ordered selection without repetition. nCrnCr Permutations: A permutation is an ordered selection without repetition. nPrnPr

16. National Lottery: Jackpot Odds 6 15 26 28 35 49 11 What would happen if you had to choose 7 balls instead of 6? 49 C 7 6 26 35 49 11 What would happen if you had to choose 5 balls instead of 6? 49 C 5 nCrnCr What about if the total number of balls was different?

17. Combinations A combination is an un-ordered selection without repetition. nCrnCr How many ways are there to choose 3 books from this shelf? How many ways are there to choose 7 cards from a suit in a deck of cards? 13 C 7 8C38C3

18. How many ways are there to choose 10 letters from the Alphabet? A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A combination is an un-ordered selection without repetition. nCrnCr 26 C 10 How many ways are there for a football manager to select a team of 11 players from the above squad 15 C 11 Combinations