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Pascal's Triangle for High School Students

        The ideas found in Pascal's Triangle can solve many problems in the real world, and also many of the problems that high school students face in the curriculum.  The binomial coefficients, which can be found on most scientific calculators with a button or formula labeled nCr, are a central concept in the field of probability, and also hold a very important place in statistics and algebra, and a clear understanding of Pascal's Triangle can make many of the problems on the SAT and other standardized tests much easier.
 
         Algebra:  The name binomial coefficients comes from algebra; if we count the top "1" in Pascal's Triangle as Row 0, then the coefficients of (a+b)ⁿ are the number in the n^th row.  For example, (x+y)⁴ = 1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴, and the numbers  1  4  6  4  1 are the 4^th row of Pascal's Triangle.  The additive rule of Pascal's Triangle can be shown by turing (a+b)ⁿ⁺¹ into (a+b)ⁿ(a+b) when (a+b)ⁿ is expanded using the n^th row of Pascal's Triangle.  The Triangle can also be used for quick ways to expand powers of 11 and powers of 9.

        Probability:  There are many discussions of probability problems on this website.  Here are some links.

How many different poker hands are there?	What is the Game of Points?
How many free throws will a 70% shooter make in 10 attempts?	What other questions are answered by Pascal's Triangle?

        Statistics:  The idea of normal distribution and the rows of Pascal's Triangle are closely related.  This Java applet can show you how closely the area under the bell-shaped curve can be approximated by the n^th row of Pascal's Triangle when all the entries are divided by 2ⁿ.  There is also a Java applet that discusses the idea of the null hypothesis H₀ and when an experiment result can be called statistically significant or highly statistically significant.