Year 8: Combinatorics Dr J FrostLast modified: 29 th December 2015 PART 1 :: Systematic Counting Sometimes the easiest way to count things is to explicitly list out every possibility! The key is listing the possibilities in some systematic way that avoids missing out cases or duplicates. Starter Hint: Perhaps consider each possible shape of triangle and how many there are of each? 14 possibilities. By breaking the problem down into types of triangle, it made it much easier for us to count without missing out any possibilities. It also prevented us having to draw out similar cases. ? Further Example A frog has 6 lily pads in front of him in a line. He can either hop (H) to the next lily pad, or skip one and go to the next (S). List out (using sequences of S and H) all the ways for the frog to get to the final lily pad. e.g. HSSH. Easiest when we break down into different numbers of skips: 0 skips:HHHHHH (1 way) 1 skip: SHHHH, HSHHH, HHSHH, HHHSH, HHHHS (5 ways) 2 skips:SSHH, SHSH, SHHS, HSSH, HSHS, HHSS (6 ways) 3 skips:SSS (1 way) 13 ways. To ensure I got all the possibilities here, I gradually moved the second S right one, before moving the left S right and repeating: SSHHHSSHHHSS SHSHHSHS SHHS In summary, break the problem down into smaller problems where its easy to count within each one, e.g. types of triangles, team sizes, number of skips, etc. ? Exercise 1 At a restaurant, there is a choice of Avocado (A), Beans (B) or Cauliflower (C) for starter, and a choice of Dog (D), Escalopes (E) and Frog Legs (F) for main course. List out all the nine possible combinations of starter and main course (e.g. BF). Solution: AD, AE, AF, BD, BE, BF, CD, CE, CF. [JMC 2006 Q4] How many triangles of any size are there in this diagram? A 8B 10C 12 D 14E 16 Solution: C [JMC 2013 Q12] How many hexagons are there in the diagram? A 4B 6C 8 D 10E 12 Solution: E 1 2 Exercises on provided sheet. 3 4 [SMC 2001 Q3] The diagram shows a regular hexagon divided up into six equilateral triangles. How many quadrilaterals are there in the diagram? A 6B 8C 10 D 12E 14 Solution: D [JMC 2006 Q17] In how many different ways can a row of five on/off switches be set so that no two adjacent switches are in the off position? A 5B 10C 11 D 13E 15 Solution: D 5 ? ? ? ? ? Exercise 1 Exercises on provided sheet [Kangaroo Pink 2005 Q8] In the diagram there are 7 squares. What is the difference between the number of triangles and the number of squares in the diagram? A 0B 1C 2 D 3E 4 Solution: D [JMO 2011 A6] The diagram shows a grid of 16 identical equilateral triangles. How many different rhombuses are there made up of two adjacent small triangles? Solution: 18 [TMC Regional 2008 Q3] In total how many triangles of any size are there in the diagram? Solution: ? ? ? ? ? Exercise 1 Exercises on provided sheet. 11 Each hour a pirate ship can either sail 1km West (W), 1km East (E), 1km North (N) or 1km South (S). It must always be moving. List all the ways in which the ship can end up 2km North after 4 hours (e.g. NNWE) and count the number of possible journeys. (Hint: it may help to break the problem down possible sets of four movements, e.g. a N, N, W and E, and then considering the possible orderings of each possible set). Solution: NNWE, NNEW, NWNE, NENW, NWEN, NEWN, WNNE, ENNW, WNEN, ENWN, WENN, EWNN, SNNN, NSNN, NNSN, NNNS (16 ways). [TMC Regional 2014 Q4] Every day Keith has a breakfast, a lunch and a dinner. The options for each meal are: One day Keith eats four items. In how many different ways can he do this? Solution: 12 [JMO 2004 A10] The Famous Five have been given 20 sweets as a reward for solving a tricky crime. They have agreed that the oldest of them must receive more than the next oldest, who must receive more than the next oldest, and so on. Assuming that each of the five gets at least one sweet, in how many different ways can they share the sweets? Solution: ? ? ? Exercise 1 Exercises on provided sheet. 14 [Cayley 2007 Q4] How many right-angled triangles can be made by joining three vertices of a cube? (This time triangles formed may also go inside the cube) Solution: 48 (In a previous question we found 24 triangles on the faces. A further 24 triangles can be formed, 2 for each face diagonal where the triangle will pass inside the cube, e.g:). There are 5 people in a room. How many ways are there of forming two teams of people, where each team must have at least one person. Solution: 15 teams. (5 teams where the teams are split 1 and teams where the teams are split 2 and 3) ? ? ? PART 2 :: Counting by Multiplying Starter: Suppose the alphabet was limited to just A, B, C, D. Fold your provided piece of paper into 16 parts. Put A 4 times, B 4 times, C 4 times and D 4 times, before cutting. Use these cards (or otherwise) to answer the following questions: Note: a word is any possible sequence of characters; it need not be in a dictionary. The number of possible 2 letter words where the letters are different. Question 1 ? The number of possible 2 letter words where the letters are unrestricted. Question 2 The number of possible 3 letter words where each letter cannot be the same as the previous letter, but is otherwise unrestricted. Question 3 The number of 4 letters words using each of the letters A, B, C, D exactly once. Question 4 The number of 5 letter words using the letters AABCD, each exactly once. Question ? ? ? ? The number of 10 letter words using the letters AAAAAABBCD each once. Question ? Multiplying and Factorial Function If the number of choices for each thing are independent of each other, then we can multiply them together to get the total number of combinations. ? ? ? Examples Q Q Q ? ? ? A Few More Examples ? ? ? Q Q Q Test Your Understanding A pack of cards contains 52 cards, 13 of each of four suits (Hearts, Spades, Clubs, Diamonds). In each suit we have the cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. Determine the number of possibilities when: 1 2 a b c d e ? ? ? ? ? ? ? ? a b c Exercise Exercises on provided sheet. ? ? ? ? ? ? ? ? ? ? ? ? Exercise 2 Exercises on provided sheet ? ? ? Exercise 2 Exercises on provided sheet ? ? Exercise 2 Exercises on provided sheet. 12 ? The Choose Function How many ways are there of selecting 2 balls from the 5? (such that the order of the balls in my selection does not matter) ? Practise using the formula (and your calculator) What would we expect this to be? (The number of ways of choosing 1 item from 10) ? What would we expect this to be? (The number of ways of choosing 0 items from 4) ? ? Examples In the UK National Lottery, you pick 6 numbers out of 59. How many possible lottery tickets are there? Q Bob the Teacher needs to form a maths team for a competition. He needs 3 boys and 3 girls for the team. He has 10 boys and 10 girls to choose from. How many possible teams are there? Q ? ? Test Your Understanding Bob the Football Manager needs to select 11 players from his squad of 15 players. How many possible selections are there? A ? I have a bowl of 5 pieces of fruit, containing an orange, apple, banana, grapefruit and kiwi. I also have in my cupboard 4 chocolate bars: a Kit Kat, Snickers, Mars Bar and Twix. If I want two pieces of fruit and two chocolate bars for my lunch, how many possible selections are there? B ? In this grid of 16 squares, I choose 4 to colour red followed by 4 to colour yellow. Determine the number of possible colourings. ? Exercise 2 Exercises on provided sheet ? ? ? ? ? ? ? ? Exercise 2 Exercises on provided sheet. 89 Start Finish ? ? ? ? Exercise 2 Exercises on provided sheet. ? Instructions: You may work in pairs. Start on the Level 1 problems provided, recording your answer on this sheet. Answers to Level 1 questions are at the front of the class. Once youve checked your answers, go to your teacher and ask for Level 2. You will need to ask the teacher to check your answers to Level 2 and beyond. Solutions are on the next slides. Levelled Activity Level 1 Level 2 Level 3 (Teacher Note: See printout) Level 1 Solutions ? ? ? ? ? ? ? ? Level 2 Solutions ? ? ? ? ? Level 3 Solutions [SMC 2005 Q16] A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are 4 substitutes: 1 goalkeeper, 1 defender, 1 midfielders and 1 forward. A substitute may only replace a player of the same category, e.g. midfielder for midfielders. Given that a maximum of 3 substitutes may be used and that there are still 11 players on the pitch at the end, how many different teams could finish the game? A 110B 118C 121 D 125E 132 Solution: B ? ? ? Level 3 Solutions [JMO 2004 A10] The Famous Five have been given 20 sweets as a reward for solving a tricky crime. They have agreed that the oldest of them must receive more than the next oldest, who must receive more than the next oldest, and so on. Assuming that each of the five gets at least one sweet, in how many different ways can they share the sweets? Solution: 7 [JMO 2002 B1] A number like 4679 is called an ascending number because each digit in the number is larger than the preceding one. (i) How many ascending numbers are there between 1000 and 2000? (ii) How many ascending numbers are there between 1000 and 10000? Solution: 56 and ? ? ? Level 3 Solutions 7 ?