5.2 Pascal’s Triangle & Binomial Theorem
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Transcript of 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
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
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
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.
5.2 Pascal’s Triangle & Binomial Theorem Amazing patterns in the
triangle…
One such pattern…
25
24
106414
or
5.2 Pascal’s Triangle & Binomial Theorem Leads to Pascal’s
Identity:Proof (see textbook):
11
1 rn
rn
rn
5.2 Pascal’s Triangle & Binomial TheoremApplications of Pascal’s Identity
712
612
713
23 b
ab
a
21
ba
116
217
216
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
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
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
5.2 Pascal’s Triangle & Binomial Theorem Home Entertainment
P289 #1,2,7,8,9,11,12,17,22