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Counting

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• 1. COUNTING
• 2. OBJECTIVES
• To be able to understand the Fundamental Principle of Counting;
• To know and to learn the different methods of counting and
• To be able to apply what are learned about Fundamental Principle of Counting.
• 3. COUNTING
• The first mathematical activity that each of us ever learned in childhood is that of counting. Skill in the simple counting processes build up as one progresses in school. This lesson deals with every useful principle in mathematics, the fundamental counting principle.
• 4. THE FUNDAMENTAL PRINCIPLE OF COUNTING Let us help the student who is taking a true-or-false test find the different patterns in answering the ten questions. Before trying to answer these questions, let us consider first a much simpler one. Instead of considering the ten questions, let us limit ourselves to just three questions. In how many ways can the three questions be answered? The different ways of answering them are shown on the diagram. It is called tree diagram because it consists of clusters of line segments or branches.
• 5. EXAMPLE 1
• 6.
• 7. The diagram shows that there are eight ways in which the three questions can be answered. Examining the diagram, we can arrive at the answer by multiplying the number of ways of answering the first question, 2, by the number of ways of answering the second question, 2, by the number of ways of asnwering the third question, 2: 2 x 2 x 2 = 8
• 8. The tree diagram method can be applied to all problems, but it is very time-consuming and impractical if we are dealing with a series of decisions, each of which contains numerous choices. Going back to our questions for a true-or-false test of ten questions, we can obtain the number of ways of answering it by following the same procedure. 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x2x2= 1 024 There are 1 024 ways .
• 9. EXAMPLE 2
• 10.
• A certain tourist wants to go to Banaue to personally see the rice terraces. There are three routes from Manila to Baguio and two routes from Baguio to Banaue.
• Look at the diagram below and see the number of ways the tourist may use to arrive at Banaue from Manila.
• 11.
• A table can also be prepared for this purpose.
• There are six ways which the tourist may take to reach Banaue from Manila.
• From the above table, the tourist may take R1 from Manila to Baguio and R4 from Baguio to Banaue, or he may take R3 from Manila to Baguio and R5 from Baguio to Banaue.
• Using the table, you can easily intrepret the other ways by which the tourist can arrive at Banaue from Manila.
Routes R4 R5 R1 (R1,R4) (R1,R5) R2 (R2, R4) (R2, R5) R3 (R3,R4) (R3, R5)
• 12.
• 13.
• LISTING METHOD
• Take note that the routes from Manila to Baguio are identified b the numbers 1,2, and 3 and the routes from Baguio to Banaue are identifies by the number 4 and 5
• A systematic listing will result to these number pairs:
• (1, 4) (2,4) (3, 4)
• (1, 4) (2,4) (3, 4)
• There are certainly 6 ways of arriving at Banaue from Manila.
• 14.
• DICE METHOD
• If a die is rolled, there are 6 ways for a number to come up: namely 1,2,3,4,5,6.
• Suppose two dice are rolled, one red and one green. How many ways can a pair of number in two colors come up.
• Since each die has six faces and there are two dice, there are 6x6 or 36 possible outcomes in all.
R/G 1 2 3 4 5 6 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
• 15. The grid table shows a pairing of numbers from the 2 dice. For example, (1,1) means that the first number,1, is the number on the red die and the second number,1, is the number on the green die; (4,5) means that the first number, 4, is the number on the red die and the second number, 5, is the number on the green die. Notice that these pairs of number are the intersections of a red die and a green die. If the pairs in the list are counted, 36 pairs (6x6 array of number pairs) consisting of red and green dice can be identified. Can you try apply this method on the example 2?
• 16. GENERALIZATION
• The generalization of this method of multiplication is called the FUNDAMENTAL PRINCIPLE OF COUNTING which states that ‘if one thing can occur in m ways and a second thing can occur in n ways, and s third thing can occur in r ways, and so on, then the sequence of things can occur in
• m x n x r x …. Ways.
• 17. EXERCISES Answer the following problems using our chosen method of counting.
• 18.
• If you buy two pairs of pants, four shirts and two pairs of shoes, how many new outfits consisting of a new pair of pants, one shirt and pair of shoes would you have?
• Ronald is taking a matching test in which he is supposed to match four answers with four questions. In how many different ways can he answer the four questions?
• A die is rolled and a coin is tossed. Determine the number of different possible outcomes by using the fundamental principle of counting. List all the possible outcomes by constructing a tree diagram.
• 19.
• 20.
• 4. A plate number is made up of two consonants followed by three nonzero digits followed by a vowel. How many plate number are possible if:
• a. The letters and digits cannot be repeated on the same plate number?
• b. The letters and digits can be repeated in the same plate number?
• 5. Three cards are drawn in succession and without replacement from a deck of cards.
• a. Find out how many ways we can obtain the king of hearts, the ace of diamonds and the ace of spades in that order.
• b. Fin the total number of ways in which the three cards can be dealt.
• 21. ANSWER KEY: (THERE CAN BE DIFFERENT METHODS USED AND IT CAN BE CONSIDERED CORRECT IF IT ARRIVED THE SAME WITH THE NUMBER OF WAYS MENTIONED BELOW:
• 16 possible outfits
• 24 ways
• 12 outcomes
• a. 1 058 400 possible plate numbers; b. 1 607 445
• a. 1 way; b. 132 600 ways
• 22. EVALUATION: ANSWER THE FOLLOWING
• How many three-digit even numbers can be formed with the digits 2,4,5,3 and 7 with no repetitions allowed?
• There are 5 roads from city A to city B and 3 roads from city B to city C. How many routes are there from city A to city C via city B?
• A haberdashery story has 6 different styles of Barong Tagalog on display. Each style is available in 2 colors. If you choose a style, you can have one of two colors. How many styles of a different color can you select from?
• 23.
• How many kinds of sandwiched can you make if you have four fillings (eggs, ham, chicken, sausage) and two choices of bread (loaf bread, rolls)?
• Going to school, Rose can take any 3 possible routes to the jeepney stop and from there take any 2 jeepney routes. How many possible ways can she have in going to school?
• 24.