Pascal’s Triangle and the Binomial Theorem Chapter 5.2 – Probability Distributions and Predictions Mathematics of Data Management (Nelson) MDM 4U

Pascal’s Triangle outer values: always 1 inner values: add the two values diagonally above

Pascal’s Triangle – the counting shortcut sum of each row is a power of 2 1 = = = = = = = 2 6

Pascal’s Triangle – Why?  Combinations!  e.g. choose 2 items from 5  go to the 5 th row, the 2 nd number = 10 (always start counting at 0)  Binomial Theorem  Patterns  Physical applications  modeling the electrons in each shell of an atom (google ‘Pascal’s Triangle electron’)

Pascal’s Triangle – Cool Stuff each diagonal is summed below and to the left called the “hockey stick” property

Pascal’s Triangle – Cool Stuff e.g., numbers divisible by 5 similar patterns for other numbers org/interactivate/ac tivities/pascal1/ org/interactivate/ac tivities/pascal1/

Pascal’s Triangle can also be seen in terms of combinations n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6

Pascal’s Triangle - Summary symmetrical down the middle outside number is always 1 second diagonal values match the row numbers sum of each row is a power of 2  sum of nth row is 2 n  Begin count at 0 number inside a row is the sum of the two numbers above it

The Binomial Theorem the term (a + b) can be expanded:  (a + b) 0 = 1  (a + b) 1 = a + b  (a + b) 2 = a 2 + 2ab + b 2  (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3  (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4  Blaise Pascal (for whom the Pascal computer language is named) noted that there are patterns of expansion, and from this he developed what we now know as Pascal’s Triangle. He also invented the second mechanical calculator.

So what does this have to do with the Binomial Theorem? remember that the binomial expansion:  (a + b) 4 = a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4  and the triangle’s 4 th row is Pascal’s Triangle allows you to determine the coefficients The exponents on the variables form a predictable pattern  The exponents of each term sum to n

The Binomial Theorem

A Binomial Expansion Expand (x + y) 4

Another Binomial Expansion Expand (a – 4) 5

Some Binomial Examples what is the 6 th term in (a + b) 9 ? don’t forget that when you find the 6 th term, r = 5 what is the 11 th term of (2x + 4) 12

Look at the triangle in a different way r0 r1 r2 r3 r4 r5 n = 0 1 n = n = n = n = n = n = for a binomial expansion of (a + b) 5, the term for r = 3 has a coefficient of 10

And one more thing… remember that for the inner numbers in the triangle, any number is the sum of the two numbers above it for example = 10 this suggests the following: which is an example of Pascal’s Identity

For Example…

How can this help us solve our original problem? so by overlaying Pascal’s Triangle over the grid we can see that there are 126 ways to move from one corner to another

How many routes pass through the green square? to get to the green square, there are C(4,2) ways (6 ways) to get to the end from the green square there are C(5,3) ways (10 ways) in total there are 60 ways

How many routes do not pass through the green square? there are 60 ways that pass through the green square there are C(9,5) or 126 ways in total then there must be 126 – 60 = 66 paths that do not pass through the green square

MSIP/ Homework Read the examples on pages , in particular the example starting on the bottom of page 287 is important Complete p. 289 #1, 2aceg, 3, 4, 5, 6, 8, 9, 11, 13