(for Teacher2Teacher Service)
Date: Jan 12, 1999 at 14:33:49
Subject: Re: Pascal's triangle
When I was reading your problem I was thinking of the typical way to introduce probability. I think that the students will understand this. However, Ido not have an answer for the one at the top of the chart which is the number of ways from among none to choose none. In high school I always answer that question with: "one way! you cannot do it!" Have the students toss pennies and keep a record of the different ways that the pennies land, throwing them one at a time. For tossing one penny, the outcome is either a H(ead) or a T(ail). Sometimes this is called a "not H(ead)!" I can imagine 9 and 10 years old kids having fun with that remark: you're a "not head!" Therefore, for the second line of Pascal's triangle: 1 1, the outcome is C(1,1) or from among one toss, there is one way to choose one head. C(1,0) =1 from among one toss there is one way to not have a head. For the third line of Pascal's triangle: 1 2 1, toss two pennies one after the other and record the outcome. The students can do this until they see that there are four possible ordered outcomes: HH, HT, TH, and TT. C(2,2) is from among two tosses there is one way to have two heads. C(2,1) is from among two tosses there are two ways to have a H and a T. C(2,0) is from among two tosses there is one way to have no heads. Similarily, you find the fourth line of the triangle 1 3 3 1. Toss three pennies and see that there is a total of 8 possible ordered outcomes. HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT. C(3,3) is from among three tosses and there is one way to have three heads. C(3,2) is from among three tosses and there are three ways to have two heads, etc. The same problem can be given with chips that are different colors on each side such as Red and White. What I believe is important for the students to understand is that HTH is not the same as THH. Hence, toss the coins one at a time not all at once. I hope that this has given you at least one idea.
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