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SLIDE 1 Find all the ways of making 60 pence using only 20 pence and 10 pence coins. Complete the table Number of 20p coins Number of 10p coins Sum 3 0 20p + 20p + 20p = 60p SLIDE 2 Number of 20p coins Number of 10p coins Sum 3 0 20p + 20p + 20p = 60p 2 2 20p + 20p + 10p + 10p = 60p 1 4 0 6 SLIDE 1 B Find all the ways of making 50 cents using only 25 cent, 10 cent and 5 cent coins. Complete the table. Number of 25� coins Number of 10� coins Number of 5� coins Sum 2 0 0 25� + 25� SLIDE 2 B Number of 25� coins Number of 10� coins Number of 5� coins Sum 2 0 0 25� + 25� 1 2 1 25� + 10� + 10� + 5� 1 1 3 25� + 10� + 5� + 5� + 5� 0 5 0 5 � 10� 0 4 2 4 � 10� + 2 � 5� 0 3 4 3 � 10� + 4 � 5� 0 2 6 2 � 10� + 6 � 5� 0 1 8 10� + 8 � 5� 0 0 10 10 � 5� SLIDE 3 Work out every possible total that can be made with one 5p, one 2p and one 1p coin using three, two or one coins SLIDE 4 5p 2p 1p Sum Coins 1 1 1 3 1 1 0 2 1 0 1 2 1 0 0 1 0 1 1 2 0 1 0 1 0 0 1 1 SLIDE 3B Work out every possible total that can be made with one 10�, one 5� and one 1� coin using three, two or one coins. SLIDE 4B 10� 5� 1� Sum Coins 1 1 1 3 1 1 0 2 1 0 1 2 1 0 0 1 0 1 1 2 0 1 0 1 0 0 1 1 SLIDE 5 The sum of 8p can be made with 5p, 2p and 1p coins in seven different ways. Complete the table to show all of these. Number of 5p coins Number of 2p coins Number of 1p coins 1 1 1 SLIDE 6 Number of 5p coins Number of 2p coins Number of 1p coins 1 1 1 1 0 3 0 1 6 0 2 4 0 3 2 0 4 0 0 0 8 SLIDE 5B The sum of 20� can be made with 10�, 5� and 1� coins in seven different ways. Complete the table to show all of these. Number of 10� coins Number of 5� coins Number of 1� coins 2 0 0 SLIDE 6B Number of 10� coins Number of 5� coins Number of 1� coins 2 0 0 1 2 0 1 1 5 0 4 0 0 3 5 0 2 10 0 1 15 0 0 20 SLIDE 7 Two cards from a pack of cards are chosen. These cards can be arranged in two different ways We count these as different arrangements There are six different arrangements of three different cards Find all six arrangements Such arrangements are also called permutations SLIDE 8 SLIDE 9 Find all the permutations of the letters A B C SLIDE 10 A B C B A C C A B A C B B C A C B A possibilities This is exactly the same question as the previous question, only using letters A B C instead of playing cards. Tutor note The permutations of three objects is a standard problem, and one that the student is required to solve with any three different items. SLIDE 11 Find all possible arrangements of the numbers 1 2 3 SLIDE 12 1 2 3 2 1 3 3 1 2 1 3 2 2 3 1 3 2 1 This is exactly the same question as the previous question, only using the numbers 1 2 3 instead of the letters A B C. SLIDE 13 Find all the permutations of the letters A B C D SLIDE 14 A B C D B A C D C A B D D A B C A B D C B A D C C A D B D A C B A C B D B C A D C B A D D B A C A C D B B C D A C B D A D B C A A D B C B D A C C D A B D C A B A D C B B D C A C D B A D C B A There are permutations SLIDE 15 The meaning of the number 4578.63 is found from the place-value of each of its digits. digit place-value number 4 1000 4000 5 100 500 7 10 70 8 1 8 units 6 4 SLIDE 16 Complete the following for the number 392.547 digit place-value number 3 9 2 5 4 7 SLIDE 17 digit place-value number 3 100 300 9 10 90 2 1 2 units 5 4 7 SLIDE 18 A number is represented by a series of digits in which each digit takes the value from 0 to 9 Complete the following ? ? SLIDE 19 ? ? SLIDE 20 We represent numbers using digits In the denary system we use the digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 There are ten digits. In counting, when we get to ten items, instead of inventing a new digit, we start over. Twelve is represented by the digits 12 where the place-value of the first 1 is ten and the place-value of the 2 is two units. ? How many units does the 3 in 325 represent? ? How many units does the 6 in 876 represent? ? How many units does the 0 in 302 represent? SLIDE 21 ? In 325 the 3 represents three hundred units ? In 876 the 6 represents six units ? The 0 in 302 represents zero tens of units SLIDE 22 In the binary system we have only two digits, 0 and 1 When we add 1 to 1 in the denary system, we get 2, but in the binary system we start over with a new digit The first four binary numbers binary denary 0 0 1 1 10 2 11 3 ? Complete the following denary binary ? The binary for the number four is 100. Complete the following binary denary 100 4 101 6 111 ? What is binary for the number eight? SLIDE 23 ? denary binary ? The binary for the number four is 100. Complete the following binary denary 100 4 101 5 110 6 111 7 ? Binary for 8 is 1000. SLIDE 24 binary denary power of 2 exponent 0 0 0 10 1 100 2 1000 8 10000 100000 1000000 64 6 SLIDE 25 binary denary power of 2 exponent 0 0 0 10 2 1 100 4 2 1000 8 3 10000 16 4 100000 32 5 1000000 64 6 SLIDE 26 Convert the following numbers from binary to denary ? 1011 ? 1100 ? 10101 ? 110001 ? 101110 SLIDE 27 ? ? ? ? ? SLIDE 28 Write the following denary numbers in binary ? 9 ? 13 ? 27 ? 32 ? 47 SLIDE 29 ? ? ? ? ? SLIDE 30 Convert the following sums in denary into binary sums ? ? ? ? ? SLIDE 31 ? ? ? ? ? SLIDE 32 Complete the following SLIDE 33 SLIDE 34 When the machine finishes turning each box can contain only one of the three pictures ? Give three different arrangements (permutations) of the three pictures ? How many different permutations are there altogether? ? You win if you have three smiley faces If each permutation is equally likely, what is the probability of your winning? SLIDE 35 ? There are twenty-seven possible solutions. For example ? There are 3 options for the first box, 3 for the second and 3 for the third. permutations in all ? Only one way of winning in 27 equally likely outcomes. Probability SLIDE 36 There are six policemen standing between traffic cones along a street while a parade passes by. If each policeman is standing between two traffic cones, how many traffic cones are there? SLIDE 37 The traffic cone question Let ?be a cone, and ?be a policeman ????????????? There are six policemen but seven traffic cones In such questions there is one more boundary than there is box SLIDE 38 A chessboard has eight by eight squares, or sixty-four squares in all. How many lines are required to draw a chessboard? SLIDE 39 There are eight squares in a row by nine edges. Same for the columns. lines are required to draw a chessboard. SLIDE 40 ? A raffle ticket contains a 4-digit number. This number cannot begin with a zero. Only one ticket will be picked to win the prize. What is the probability of winning the prize? ? A telephone book contains 5-digit numbers not beginning with a 0 or 1, and not ending in a 0. How many possible telephone numbers are there? SLIDE 41 ? A raffle ticket contains a 4-digit number. This number cannot begin with a zero. Only one ticket will be picked to win the prize. What is the probability of winning the prize? There are 9000 numbers between 1000 and 9999 Probability ? A telephone book contains 5-digit numbers not beginning with a 0 or 1, and not ending in a 0. How many possible telephone numbers are there? There are 8 ways of picking the first digit, 10 ways of picking the second digit, 10 ways of picking the third digit, 10 ways of picking the fourth digit and 9 ways of picking the last digit. SLIDE 42 A class contains 13 students. Why must two of these students have a birthday in the same month? SLIDE 43 A class contains 13 students. Why must two of these students have a birthday in the same month? Solution There are 12 months in a year, but here there are 13 students. Suppose 12 of these students have birthdays in different months. Then the last student must have a birthday in the a month that is the same as one of the other students. Note This is an application of the Dirichlet pigeon-hole principle. For example, you cannot fit seven pigeons into six pigeon-holes. SLIDE 44 A chessboard has sixty-four squares. Two squares are removed as shown in diagram A, leaving sixty-two squares. You have 31 domino pieces, as shown in the diagram. You are instructed to cover the sixty-two squares of the board so that the dark side of the domino covers a dark square on the board, and the light side covers a white square. In figure 2 two domino pieces have been placed on the board. Is it possible to do this? Explain your answer. SLIDE 45 A chessboard has sixty-four squares. Two squares are removed as shown in diagram A, leaving sixty-two squares. You have 31 domino pieces, as shown in the diagram. You are instructed to cover the sixty-two squares of the board so that the dark side of the domino covers a dark square on the board, and the light side covers a white square. In figure 2 two domino pieces have been placed on the board. Is it possible to do this? Explain your answer. This is not possible. The two squares that have been removed from the chessboard are both white squares. There are 32 dark squares and 30 white squares, but in the collection of 31 dominos there are 31 dark squares and 31 light squares. So the board cannot be covered in this way. SLIDE 46 The diagram shows how a knight in chess can move. Find all the squares that a knight could move to in one and two moves from the square shown in the diagram. SLIDE 47 SLIDE 48 The diagram shows a network of roads between four towns with distances in miles. Find all the possible ways of going from A to D and the length of these routes. Which route is the shortest? SLIDE 49 Route ABD Length Route AD Length 31 miles Route ACD Length Shortest route: ACD SLIDE 50 The diagram shows a network of roads between four towns with distances in miles. Find all the possible ways of going from A to E and the length of these routes. Which route is the shortest? SLIDE 51 Route ABE Length Route ADE Length Route ACE Length Route ADCE Length Route ACDE Length Shortest route: ADCE SLIDE 52 The diagram shows a network of roads between four towns with distances in miles. Find all the possible ways of going from A to E and the length of these routes. Which route is the shortest? SLIDE 53 Route ABE Length Route ADE Length Route ACE Length Route ADBE Length Route ABDE Length Route ADCE Length Route ACDE Length There are two shortest routes: ADE and ABDE

Combinations

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