11. Combinations and Permutations PROJECT 11.3 Pascal’s Triangle – first 20 lines


Write out the first 20 lines of Pascal’s Triangle. Then on a blank sheet draw a triangle with 20 lines marked on it and one space left for each of the numbers of Pascal’s Triangle. Instead of writing all the numbers just shade in the spaces which contain even numbers. Start like this:


1 1 1 1 1


1 1 2


3 1


4 64 3


5 10 10 1 1 5 1


When you have completed Chapter 13, Chaos, you should check this triangle. After completing 20 lines, you may recognise the shape!


Pascal’s Triangle arises in connection with the Binomial Theorem in algebra. It makes it easy to multiply out expressions like (x + y)4


. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1


. . . (x + y)1 . . . (x + y)2 . . . (x + y)3 . . . (x + y)4


= x + y = x2 = x3 = x4


+ 2xy + y2 + 3x2 + 4x3


y1 y1


+ 3x1 + 6x2


y2 y2


+ y3 + 4x1 y3 + y4


Example 6:


Expand (x + y)6 = x6


using Pascal’s Triangle. The relevant


line of Pascal’s Triangle is 1, 6, 15, 20, 15, 6, 1. (x + y)6


+ 6x5 y1 + 15x4 y2 + 20x3 y3 + 15x2 y4 This would take a long time to multiply out. + 6x1 y5 + y6


109


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