5.2 Pascal’s Triangle & Binomial Theorem

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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

description

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

Transcript of 5.2 Pascal’s Triangle & Binomial Theorem

Page 1: 5.2 Pascal’s Triangle & Binomial Theorem

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

row

diag

onal

Page 2: 5.2 Pascal’s Triangle & Binomial Theorem

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 #

Page 3: 5.2 Pascal’s Triangle & Binomial Theorem

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…

Page 4: 5.2 Pascal’s Triangle & Binomial Theorem

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?

Page 5: 5.2 Pascal’s Triangle & Binomial Theorem

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?

Page 6: 5.2 Pascal’s Triangle & Binomial Theorem

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

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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

Page 8: 5.2 Pascal’s Triangle & Binomial Theorem

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

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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

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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

3. Routes that avoid C and get to B.

210610

)6,10(

C

434

)3,4(

C

2036

)3,6(

C

13020421036

34

610

8020436

34

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5.2 Pascal’s Triangle & Binomial TheoremPascal’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.

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5.2 Pascal’s Triangle & Binomial Theorem Amazing patterns in the

triangle…

One such pattern…

25

24

106414

or

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5.2 Pascal’s Triangle & Binomial Theorem Leads to Pascal’s

Identity:Proof (see textbook):

11

1 rn

rn

rn

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5.2 Pascal’s Triangle & Binomial TheoremApplications of Pascal’s Identity

712

612

713

23 b

ab

a

21

ba

116

217

216

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5.2 Pascal’s Triangle & Binomial Theorem How many routes are possible for the checker to

finish at the bottom right corner?

1

1 1

1 2 1

1 3 3 1

1 4 6 4

5 10 10 4

5 15 20 14

20 35 34 14

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5.2 Pascal’s Triangle & Binomial Theorem

A Few Things To A Few Things To Note: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…

1

nn

10

n

nn

1

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5.2 Pascal’s Triangle & Binomial Theorem

MORE…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.)

rn

nrn

n

nn

nnnnn

21

...210

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