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.
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.
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
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 .
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?
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.
EXERCISES Answer the following problems using our chosen method of counting.
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?
There are 3 commuter trains and 4 express buses departing from town P to town Q in the morning and 2 commuter trains and 3 express buses operating on the return trip in the evening. In how many ways can a commuter from town P to town Q complete a daily round trip via bus and/or train?
Try Me Burger shop offers a combination consisting a cup of soup, sandwich and beverage at a special price. There two kinds of soups (corn and asparagus), three sandwiches (chicken, ham and mushroom) and three beverages (coffee, tea and milk) to choose from.
A. Determine how many different meal combination are possible.
B. Construct a tree diagram to list all possible combinations.