5.2 Pascal’s Triangle & Binomial Theorem 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… http:// mathforum.org/workshops/usi/pascal/mo.pascal.html row diagonal
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… - PowerPoint PPT Presentation
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5.2 Pascal’s Triangle & Binomial TheoremConsider 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
5.2 Pascal’s Triangle & Binomial TheoremAdd 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 #
5.2 Pascal’s Triangle & Binomial TheoremPascal’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…
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.
1. How many pizzas can you order with no toppings? 2. How many pizzas can you order with all eight toppings? 3. How many pizzas can you order with only one topping? 4. How many pizzas can you order with seven toppings? 5. How many pizzas can you order with two toppings? 6. How many pizzas can you order with six toppings?
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?
5.2 Pascal’s Triangle & Binomial Theorem 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
B
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:
N E N E E N E N N E E N E E
N E E N E E N N E E N E E N
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)
3003814
614
)8,14(6,14
CC
!8!6!14
5.2 Pascal’s Triangle & Binomial TheoremExample: On a 6 by 4 grid:1. How many routes go from A(0,0) to B(6,4)?2. How many routes pass through C(3,1) to get to B?3. How many routes avoid C to get to B?
A
B
C
5.2 Pascal’s Triangle & Binomial Theorem Sol’n1. Number of routes from A to B.
2. Number of routes from A to C.Number of routes from C to B.Number of routes thru C to B