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Notes_MAT017Chapter13

# Notes_MAT017Chapter13 - Kutztown University of Pennsylvania...

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Kutztown University of Pennsylvania Department of Mathematics MAT017 – Introduction to Mathematics Spring Term 2008 Chapter 13.1 Notes on Counting In class, it was determined that there were eight (8) possible ways of having three children in the family. A list of outcomes is given by the sample space S : S ={ bbb , bbg , bgb , bgg , gbb , gbg , ggb , ggg }, and the cardinal number of set S , n ( S ) = 8. By actually writing them down, we were able to count them. Also, how about if both parents want six children and want to determine all possible outcomes? We can start by writing each outcome down as we did before, but this approach will be tedious, but nevertheless, the sample space can be found. Rather, we are going to look at a method called “Counting” to determine a systematic way of determining the number of ways this particular family can have six children. It is very fast and efficient. The Fundamental Counting Principle (FCP) “If we wish to perform a series of tasks and the first task can be done in a ways , the second can be done in b ways , the third can be done in c ways , and so on, then all tasks can be done in a x b x c x …. ways .” So, let’s investigate the original three-child problem to determine how the eight was determined through this method of FCP. Example 1 In the first child, there are 2 (which equals a ) ways (that is, sexes possible), for the second sibling, there are also 2 (which equals b ) sexes possible, and for the third, there are also 2 (which equals c ) sexes possible. So, according to the method of FCP, n ( S ) = a x b x c = 2 x 2 x 2 = 8 ways or outcomes. Example 2 For the six-child problem, using similar arguments for the three-child case, a = 2, b = 2, c = 2, d = 2, e = 2, and f = 2. So, the cardinal number for this set, say A , n ( A ) = 2 x 2 x 2 x 2 x 2 x 2 = 2 6 = 64. I guess we can list all possible outcomes, but this will be very tedious. Anyways, here are some elements/outcomes to set A : A = { bbbbbb , bbbbbg , bbbbgb , bbbgbb , …, ggggggg }. Instructor: Dr. Perry Lee 1

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Kutztown University of Pennsylvania Department of Mathematics MAT017 – Introduction to Mathematics Spring Term 2008 So, we see that there will be 64 different outcomes from a six-child family.
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