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5.2 Pascal’s Triangle & Binomial Theorem

5.2 Pascal’s Triangle & Binomial Theorem. diagonal. Consider the triangle arrangement at the right... What pattern is used to create each row? What pattern is in the 2 nd diagonal? What pattern is in the 3 rd diagonal? Check out this link…

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5.2 Pascal’s Triangle & Binomial Theorem

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  1. 5.2 Pascal’s Triangle & Binomial Theorem diagonal Consider the triangle arrangement at the right... • What pattern is used to create each row? • What pattern is in the 2nd diagonal? • What pattern is in the 3rd diagonal? • Check out this link… http://mathforum.org/workshops/usi/pascal/mo.pascal.html row

  2. 5.2 Pascal’s Triangle & Binomial Theorem Add terms in: • First row (row #0) • Second row (row #1) • Third row (row #2) • Forth row (row #3) • Fifth row (row #4) What conclusion can you make about the sum of the terms in the row and the row number? ∑=1 ∑=2 ∑=4 ∑=8 ∑=16 Sum of the row equals 2 raised to the power of that row #

  3. 5.2 Pascal’s Triangle & Binomial Theorem Pascal’s Pizza Party! • Pascal and his pals have returned home from their rugby finals and want to order a pizza. They are looking at the brochure from Pizza Pizzaz, but they cannot agree on what topping or toppings to choose for their pizza. Pascal reminds them that there are only 8 different toppings to choose from. How many different pizzas can there be? • Descarte suggested a plain pizza with no toppings, while Poisson wanted a pizza with all eight toppings. • Fermat says, “What about a pizza with extra cheese, mushrooms and pepperoni?” • Pascal decides they are getting nowhere…

  4. 5.2 Pascal’s Triangle & Binomial Theorem • Here are the toppings they can choose from: • Pepperoni, extra cheese, sausage, mushrooms, green peppers, onions, tomatoes and pineapple. • How many pizzas can you order with no toppings? • How many pizzas can you order with all eight toppings? • How many pizzas can you order with only one topping? • How many pizzas can you order with seven toppings? • How many pizzas can you order with two toppings? • How many pizzas can you order with six toppings?

  5. 5.2 Pascal’s Triangle & Binomial Theorem • Find the numbers of different pizza options in Pascal’s triangle. • Can you use Pascal’s triangle to help you find the number of pizzas that can be ordered if you wanted three, four, or five toppings on your pizza? • How many different pizzas can be ordered at Pizza Pizazz in total?

  6. 5.2 Pascal’s Triangle & Binomial Theorem B • On the Island of Manhattan in NYC, the surface streets network is set up on a rectangular grid with the Avenues running North-South and the Streets running East-West. If you took a taxi from point A to point B that traveled only North or East, how many possible routes could the driver follow? A

  7. 5.2 Pascal’s Triangle & Binomial Theorem • Sol’n • To get from A to B there is some combination of 6-north movements and 8-east movements. • To get from start to finish there are a total of 14 “blocks” to traverse. • One possible route may be: • Another:

  8. 5.2 Pascal’s Triangle & Binomial Theorem • The number of routes is equivalent to determining the number of ways N can be inserted into the 6 positions from the 14 possible (6 duplicate N, 8 duplicate E). • Using Combinations (order is unimportant)

  9. 5.2 Pascal’s Triangle & Binomial Theorem Example: On a 6 by 4 grid: • How many routes go from A(0,0) to B(6,4)? • How many routes pass through C(3,1) to get to B? • How many routes avoid C to get to B? B C A

  10. 5.2 Pascal’s Triangle & Binomial Theorem • Sol’n • Number of routes from A to B. • Number of routes from A to C. Number of routes from C to B. Number of routes thru C to B • Routes that avoid C and get to B.

  11. 5.2 Pascal’s Triangle & Binomial Theorem Pascal’s Triangle • Blaise Pascal noted a pattern in the expansions of several different powers of (a+b) {(a+b)2, (a+b)3, (a+b)4,(a+b)5,etc.} • The triangular array of coefficients of these expansions became known as Pascal’s Triangle.

  12. 5.2 Pascal’s Triangle & Binomial Theorem • Amazing patterns in the triangle… • One such pattern…

  13. 5.2 Pascal’s Triangle & Binomial Theorem • Leads to Pascal’s Identity: Proof (see textbook):

  14. 5.2 Pascal’s Triangle & Binomial Theorem Applications of Pascal’s Identity

  15. 5.2 Pascal’s Triangle & Binomial Theorem • How many routes are possible for the checker to finish at the bottom right corner?

  16. 5.2 Pascal’s Triangle & Binomial Theorem A Few Things To Note: • There is only one way to choose all of the elements… • There is only one way to choose none of the elements… • There are n ways to choose 1 element from n elements…

  17. 5.2 Pascal’s Triangle & Binomial Theorem MORE… • Choosing r elements from n elements is the same as choosing n-r to ignore… • (e.g., Choosing 3 girls from 8 girls for a committee is the same as choosing 5 girls to not be on the committee) • The number of collections of any size from n elements is… • (e.g., the number of different playlists selected from 10 tunes is 210.)

  18. 5.2 Pascal’s Triangle & Binomial Theorem • Home Entertainment P289 #1,2,7,8,9,11,12,17,22

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