1. 1 Press Ctrl-A ©G Dear2009 – Not to be sold/Free to use CountingTechniques Stage 6 - Year 12 General Mathematic (HSC)

2. 2 Counting N Objects Nn! N objects can be arranged in n! different ways. 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 5! = 5 x 4 x 3 x 2 x 1 = 120 n! Use the n! key on your calculator. How many ways can we arrange, & ? 1. 2. 3. 4. 5. 6. 3! = 3 x 2 x 1 =6

3. 3 Nb. It’s important who comes 1 st and 2 nd in a race. number of elements in the sample space multiplyingnumber of choices for 1 st placethe number of choices for 2 nd place The number of elements in the sample space can be found by multiplying the number of choices for 1 st place by the number of choices for 2 nd place. There are 5 runners (numbered 1 to 5) in a race. How many ways can we get 1 st and 2 nd place. 1. 1.1, 2 2. 2.1, 3 3. 3.1, 4 4. 4.1, 5 5. 5.2, 1 6. 6.2, 3 7. 7.2, 4 8. 8.2, 5 9. 9.3, 1 10. 10.3, 2 11. 11.3, 4 12. 12.3, 5 13. 13.4, 1 14. 14.4, 2 15. 15.4, 3 16. 16.4, 5 17. 17.5, 1 18. 18.5, 2 19. 19.5, 3 20. 20.5, 4 5 x 4 = 20 1 st,2 nd Counting Ordered Selections

4. 4 NOTwhere Nb. It’s NOT important who comes where. number of elements in the sample space dividingnumber of ordered selections number of arrangement The number of elements in the sample space is found by dividing the number of ordered selections by the number of arrangement of selection. 1. 4x312  2 = 6 How many ways can we select two colours from,, & ? 2.3.4.5.6.7.8.9.10.11.12. ????  2! = Note that some combinations repeat. Counting Unordered Selections

5. 5 5 players5 substitutes There are 5 players and 5 substitutes on a basketball team. 1.team Prime Minister 1. The team is arranged in line to meet the Prime Minister. How many ways can they be arranged? 10! = 3628800 2.5 players 2. How many ways to choose a team of 5 players? Unordered = 10 x 9 x 8 x 7 x 6  5! Hint Button = 252 3.captainvice captain 3. How many ways to choose a captain and vice captain? Ordered = 10 x 9 Hint Button = 90 Multiplynumber of choices for 1st placethe number of choices for 2nd place Multiply the number of choices for 1st place by the number of choices for 2nd place. Dividenumber of ordered selections number of arrangement Divide the number of ordered selections by the number of arrangement Ordered Unordered Counting Techniques