1
While searching for a fun and interesting lesson to start the school year, I
ran across several activities that relate to Pascal’s Triangle. I know that students
are often intimidated to start the first day of school with heavy-duty math
concepts, so this array of numbers seemed like a great place to start. The
activity I chose to use allowed students to work in groups to discover several
elementary patterns. I know that students tend to get excited when they make
discoveries so I thought it would be fun way to get started. Beginning the year
with a lesson on Pascal’s Triangle allowed me to build upon seemingly simple
patterns throughout the year.
I have always found Pascal’s Arithmetic Triangle to be fascinating. I was
aware of several useful applications that turn up in typical high school math
textbooks. Through my research I found a number of applications that I was
unaware of. What the reader will find in this paper is a collection of interesting
patterns and useful applications of Pascal’s Triangle. The number of patterns is
countless, so this paper is just the tip of the iceberg. Also included are some
student worksheets that can be used in a middle school or high school
classroom. Each worksheet will lead to discoveries by students which will allow
them to build on previous knowledge.
2
Figure 1: Pascal’s Triangle (Green, 1986)
Pascal’s Arithmetic Triangle is a triangular array of whole numbers. It is
also known as the figurate triangle, the combinatorial triangle, and the binomial
triangle. The top of the triangle always starts with the number 1. Numbers in a
horizontal line make up a row. The first and last numbers in each row are 1.
Each of the other numbers in a row is found by adding the two numbers above it.
It is generally understood that the top row, which contains a single number 1, is
row 0. This is because it corresponds to the expansion of
. Binomial
expansion will be discussed later in this paper. The rows continue to be counted
3
row 1, row 2, row 3, etc. There are an infinite number of rows because the
triangle itself is infinite. Numbers in an oblique line on the diagonal are called
diagonals.
Figure 2: Rows and Diagonals (Seymour, 1985)
A number in Pascal’s Triangle is referred to as an element. The number
of elements is always one more than the number of the row. For example, there
are 5 elements row 4. The number of elements in a diagonal is infinite because
the diagonal never ends.
Figure 3: Elements (Seymour, 3)
Although Pascal’s Triangle is named after 17th century mathematician
Blaise Pascal, many other mathematicians knew about and used their knowledge
of this array hundreds of years before the birth of Pascal. There is some proof
that this number triangle was familiar to the Arab astronomer and mathematician
4
Omar Khayyam as early as the 11th century.1 Some historians believe that the
numbers originally arose from the Hindu study of combinatorics and binomial
numbers as well as the Greeks study of figurate numbers.2 The numbers appear
to have been discovered independently by both Persian and Chinese
mathematicians in the 11th century.3
In 1261, Yang Hui listed the numbers up to row 6. He called the triangle
“the tabulation system for unlocking binomial coefficients.” The same triangle
extended to row 8 was given by Chu Shih-Chieh in 1303.4 The Chinese began
calling the numbers “Yanghui’s Triangle” because Chu Shih-Chieh gave credit to
Yanghui who worked with the triangle 40 years prior.5
Most likely the triangle came to Europe from China through Arabia.6 In
Europe, there are many authors who can lay claim to having made a serious
study of the triangle. Several of them studied the triangle long before Pascal was
even born.7 In some Italian works, the array is called “Tartaglia’s Triangle”
named after Italian algebraist Tartaglia who published the numbers in 1556.8
Other mathematicians who worked on the triangle are Gerolamo Cardano in
1
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008
<http://pages.csam.montclair.edu/~kazimir/index.html>
2
“Pascal’s Triangle-History in Europe.” Pascal’s Triangle From Top to Bottom. 4 Oct. 2008
<https://www.math.ucon.edu/~troby/hidden/4math/ptw/europe.html>.
3
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008
<http://pages.csam.montclair.edu/~kazimir/index.html>
4
A.W.F Edwards, Pascal’s Arithmetical Triangle. (London: Charles Griffin & Company LTD,
1987) 51.
5
“Pascal’s Triangle-History in Europe.” Pascal’s Triangle From Top to Bottom. 4 Oct. 2008
<https://www.math.ucon.edu/~troby/hidden/4math/ptw/europe.html>.
6
“Pascal Triangle – History” Fibonacci Numbers and the Pascal Triangle. 4 Oct. 2008
<http://milan.milanovic.org/math/english/fibo/fibo0.html>
7
“Pascal’s Triangle-History in Europe.” Pascal’s Triangle From Top to Bottom. 4 Oct. 2008
<https://www.math.ucon.edu/~troby/hidden/4math/ptw/europe.html>.
8
Kazimir, Pascal’s Triangle. 4 Oct. 2008 <http://pages.csam.montclair.edu/~kazimir/index.html>
5
1539, Francois Viete in 1591, William Oughtred in 1631, Henry Briggs in 1633
and Father Marin Mersenne in 1663.9
The numbers were published many times in countries around the world
before Pascal, but he did a lot of the original work. He also discovered many
identities in the numbers.10
Blaise Pascal (1623-1662) was part of a generation of great
mathematicians in France that include Decartes, Fermat and Father Marin
Mersenne.11 Pascal invented a calculating machine, worked on the barometer
and produced work on conic sections.12
Pascal and his father, Etinne, came in contact with Father Marin
Mersenne who published a book with a table of binomial coefficients in 1636.13
Pascal also entered into correspondence with Pierre de Fermat in 1654. This
correspondence was in regard to a problem in calculating the odds in games of
chance. This problem, the Gambler’s Problem of Points, concerned the division
of the stakes between two players when a game has to be left unfinished. This
9
Kazimir, Pascal’s Triangle. 4 Oct. 2008 <http://pages.csam.montclair.edu/~kazimir/index.html>
10
“Pascal Triangle – History” Fibonacci Numbers and the Pascal Triangle. 4 Oct. 2008
<http://milan.milanovic.org/math/english/fibo/fibo0.html>
11
“Pascal’s Triangle-History in Europe.” Pascal’s Triangle From Top to Bottom. 4 Oct. 2008
<https://www.math.ucon.edu/~troby/hidden/4math/ptw/europe.html>.
12
Tony Colledge, Pascal’s Triangle: A Teacher’s Guide with Blackline Masters (Minneapolis:
Tarquin Publications, 1997) 2.
13
“Pascal Triangle – History” Fibonacci Numbers and the Pascal Triangle. 4 Oct. 2008
<http://milan.milanovic.org/math/english/fibo/fibo0.html>
6
correspondence resulted in Pascal’s most famous publication, “Traite du Triangle
Arithmetique,” probably in August of 1654.14
In his publication, Pascal explained how the figurate numbers were in fact
both the combinatorial numbers and the binomial numbers. He also developed
many of the triangles properties and applications within these writings.
Pascal was not the first man in Europe to study the binomial coefficients.
However, Pascal’s contributions to math, especially of his triangle were
unquestionably brought forth from the mind of a highly intelligent man.15
It is considered bad form to name something after yourself.
Mathematicians name things by giving credit the author of a paper after the fact.
Monmort, writing in French in 1708 called this array of numbers the
“combinatorial table of Mr. Pascal”.16 Pascal’s extensive work on probability
theory is what caused the triangle to be named after him.17
Pascal’s Triangle contains several amazing and interesting patterns. Its
applications are incredible, as well. One of the first mysteries to be spotted is
that the numbers of the array are symmetric. A line drawn vertically through the
center of the triangle creates mirror images as seen in Figure 4.
14
“Pascal Triangle – History” Fibonacci Numbers and the Pascal Triangle. 4 Oct. 2008
<http://milan.milanovic.org/math/english/fibo/fibo0.html>
15
Kazimir, Pascal’s Triangle. 4 Oct. 2008 <http://pages.csam.montclair.edu/~kazimir/index.html>
16
Pascal’s Triangle-History in Europe.” Pascal’s Triangle From Top to Bottom. 4 Oct. 2008
<https://www.math.ucon.edu/~troby/hidden/4math/ptw/europe.html>.
17
Tony Colledge, Pascal’s Triangle: A Teacher’s Guide with Blackline Masters (Minneapolis:
Tarquin Publications, 1997) 2.
7
Figure 4: Symmetry in Pascal’s Triangle (Seymour, 2)
There are also patterns found in the diagonals of Pascal’s Triangle. To
begin, diagonal 0 is full of 1s. Diagonal 1 is the sequence of counting numbers.
Diagonal 2 is the set of triangular numbers. The successive triangular numbers
are
where
term and
. So,
stands for the
.18 In simpler terms, the triangular
numbers are 1, and
and so on, as pictured in
Figure 5.
Figure 5: The first 4 triangular numbers (Edwards, 2)
Adding any two successive numbers in diagonal 2 results in a perfect
square. A square number, , can be pictured as rows of dots and is equal to
(see Figure 6). In other words, the third perfect square, 9, can be pictured as a
square consisting of 3 rows of 3 dots. Also,
18
19
A.W.F. Edwards Pascal’s Arithmetical Triangle. (London: Charles Griffin & Company LTD,
1987) 1.
19
A.W.F. Edwards Pascal’s Arithmetical Triangle. (London: Charles Griffin & Company LTD,
1987) 1.
8
Figure 6: The first 4 square numbers (Edwards, 2)
The set of tetrahedral numbers (1, 4, 10, 20 …) is given in diagonal 3.
Tetrahedral numbers are the sum of the triangular numbers (1, 1
6, 1
4
6
3, 1
3
10 .20
In Figure 7, the numbers located on each of the drawn-in diagonals of
Pascal’s Triangle sum to the Fibonacci sequence.21 Recall that Fibonacci’s
sequence begins with two 1s and then all other numbers are generated by
summing the two numbers before the next term in the sequence (1, 1, 1
1
2
3, 12
3
1
2,
5 .
Figure 7: Fibonacci’s sequence in Pascal’s Triangle (Green, 78).
The horizontal rows represent powers of 11. Row 0 is 11
11
20
11 (1, 1). Row 2 is 11
1. Row 1 is
121 (1, 2, 1).
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008
<http://pages.csam.montclair.edu/~kazimir/index.html>
21
Thomas Green. Pascal’s Triangle. (Minneapolis: Dale Seymour Publications, 1986) 78.
9
Add the numbers in each row of Pascal’s Triangle. What pattern is found
in the results? Adding rows shows a pattern of doubling. The sums are powers
of 2. The sum of the elements in row 3 is 1
1
4
6
4
1
16
3
3
1
8
2 . In row 4,
2 .
For any prime numbered row, or row where the first element is a prime
number, all of the numbers in that row (excluding the 1s) are divisible by that
prime. In mathematical terms it can be stated that
“if is a prime number, then all the middle elements (all elements except
the two end elements) of the
row are divisible by ”.22
For example, row 7 consists of the numbers 1, 7, 21, 35, 35, 21, 7, and 1.
Besides the 1s that begin and end the row, all of the numbers are divisible by 7.
On the other hand, if
is a composite number then some elements in the
row
will be divisible by .
Another pattern found in Pascal’s Triangle is called the Hockey Stick
pattern. Sum the first four elements of diagonal 2: 1
3
6
10
20.
Outlining these numbers and the resulting sum creates a figure that is shaped
like a hockey stick as shown in Figure 8. The same can be seen when you add
the first five elements of diagonal 2. The diagonal of numbers of any length
starting with any of the 1s bordering the side of the triangle and ending on any
element inside the triangle is equal to the number below the last element of the
22
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008
<http://pages.csam.montclair.edu/~kazimir/index.html>
10
diagonal not on the diagonal.23 This is true because
1
1
as
proved in Appendix A. Because of the symmetry in Pascal’s Triangle that was
discussed earlier, the hockey stick pattern can be formed in either direction.24
Figure 8: Hockey Stick Pattern (Green, 53).
Another interesting pattern that emerges is in the products of rings in
Pascal’s Triangle. When you multiply the numbers in a ring, the product is
always a perfect square number. 25 Take one of the examples in Figure 9.
23
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008
<http://pages.csam.montclair.edu/~kazimir/index.html>
24
Thomas Green. Pascal’s Triangle (Minneapolis: Dale Seymour Publications, 1986) 53.
Dale Seymour. Visual Patterns in Pascal’s Triangle. (Minneapolish: Dale Seymour
Publications, 1985) 45
25
11
Figure 9: Rings in Pascal’s Triangle (Seymour, 45).
Multiplying the elements in the ring with 3 as a center results in 1
4
1
144
2
3
6
12 . Also, in the ring with 84 as it’s center, we will multiply 56, 28,
36,120, 210, and 126.
56 2 · 7
28 2 · 7
36 2 · 3
120 2 · 3 · 5
210 2 · 3 · 5 · 7
126 2 · 3 · 7
So, 56 · 28 · 36 · 120 · 210 · 126 2
·3 ·5 ·7
Since this result has all even powers, it is a perfect square!
A student activity and worksheet, Exploring Patterns, is provided in
Appendix B.
Some of the most interesting patterns are visual patterns that can be seen
when the elements of the triangle are colored. For example, when the even
numbers are colored a different color than the odd numbers, the result is a
12
pattern that is the same as the Sierpinski Triangle (see Figure 10). This triangle
is a fractal named after Wacław Sierpiński who described it in 1915.26 The
pattern shows stepped triangles.
Figure 10: Evens and Odds (Seymour, 19)
What’s even more interesting is that this fractal does not only show up
when evens (multiples of 2) are colored. Generating the pattern made by
coloring the multiples of other numbers is an extension of the work on odd and
even numbers. A pattern that is eerily similar to the Sierpinski triangle emerges
when multiples of 3, 4, 5, 6, 7, 8, 9, and 10 are colored as shown in Figure 11.
Figure 11: Multiples of 3, 4 and 7 (Seymour, 19).
A student activity worksheet, Coloring Multiples, is included in Appendix C.
26
“Applicaions of Pascal’s Triangle.” Learn Something New Every Day. 4 Oct. 2008
<http://mihirknows.blogspot.com/2007/10/applications-of-pascals-triangle.html>
13
While the patterns described above are fun and interesting, there are
many aspects of Pascal’s Triangle that are applicable to middle and high school
math.
One of these applications is probability. The State of Ohio Board of
Education lists Probability as one of its Academic Content Standards. A grade
level indicator for the eighth grade states that students should be able to:
“Calculate the number of possible outcomes for a situation, recognizing
and accounting for when items may occur more than once or when order
is important”.27
A grade level indicator for the ninth grade states that students should:
“Use counting techniques and the Fundamental Counting Principle to
determine the total number of possible outcomes for mathematical
situations”.28
Pascal’s Triangle can be used to allow students to demonstrate
knowledge in both of these areas. It shows you the results of heads and tails
when a fair, 2-sided coin is tossed.29 For example, let us say that a fair, 2-sided
coin is tossed three times. What is the probability that the three tosses will result
in 2 heads and 1 tail? There is only 1 way that the tosses will result in three
heads. There are 3 ways that two heads and one tail can be tossed. There are
also 3 ways that one head and two tails can be tossed. And finally, there is only
1 way that three tails can be tossed. Now, look at row 3 of the triangle. The
elements, 1, 3, 3, 1 give us the number of times each result can occur. So, it is
27
Ohio Department of Education, comp. Academic Content Standards: K-12 Mathematics.
(Columbus, OH: State of Ohio Bard of Education, 2001) 105
28
Ohio Department of Education, comp. Academic Content Standards: K-12 Mathematics.
(Columbus, OH: State of Ohio Bard of Education, 2001) 105
29
“Applicaions of Pascal’s Triangle.” Learn Something New Every Day. 4 Oct. 2008
<http://mihirknows.blogspot.com/2007/10/applications-of-pascals-triangle.html>
14
easy to see that there is a 3 out of 8 chance that tossing three coins will result in
2 heads and 1 tail. (See Coin Tossing Activity: Appendix D and Investigation:
Appendix E).
Pascal’s Triangle can also be used to teach middle and high school
students about combinations.30 The elements of Pascal’s Triangle can be used
to find out how many subsets of
elements can be formed from a set with
distinct elements.31 For example, let us say that we must choose three videos
from a box of seven. In how many different ways can we chose the videos? The
!
solution can be found using the combination rule
7 and
3. So,
7
3
!
! !
!
· · · · · ·
! !
· · · · · ·
, where in this case
! !
35.
Pascal’s Triangle saves the trouble of using this tedious formula. Since
we are choosing three elements from seven, we will find the 3rd element in row 7.
Row 7 consists of the elements 1, 7, 21, 35, 35, 21, 7, 1. Remember that the first
1 is element 0. So, the 3rd element is 35. This means in the formula
equal to the row number of the triangle where
,
is
is equal to the element in that
particular row.32 (See Appendix F: Pizza Problem).
As one familiar with Algebra may know, the numbers in each row of the
triangle are precisely the same numbers that are the coefficients of binomial
30
Prentice Hall, comp. Middle Grades Math: Tools for Success. (Upper Saddle River, New
Jersey: Prentice Hall, 1999) 553
31
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008 <http://pages.csam.montclair.edu/~kazimir/
32
Jessica Kazimir, Pascal’s Triangle. 4 Oct. 2008 <http://pages.csam.montclair.edu/~kazimir/
15
expansions.33 The Ohio State Board of Education indicates that students in the
twelfth grade should be able to:
“Apply combinations as a method to create coefficients for the Binomial
Theorem, and make connects to everyday and workplace problem
situations”.34
Consider the expansion of
. To simplify, one may multiply
. The result is
variables are all combinations of
and
4
6
4
. The
where the sums of the exponents are
equal to four. In other words, the degree of each term is four. The coefficients of
the terms (1, 4, 6, 4, 1) correspond exactly to the numbers in row 4 of Pascal’s
Triangle. So, in general, the th row of the triangle gives the expansion
coefficients of
. This is proved in Appendix G.
A direct expansion for each coefficient is
…
…
. This
is the same formula for finding combinations, as discussed previously. So, the
coefficients of any binomial expansion can be found by finding
where the
upper index, , is the exponent for the first variable and the lower index, , is the
term.35 (See Appendix H: Binomial Coefficients).
The patterns that can be found in Pascal’s Triangle are endless, so the
patterns discussed in this paper are just the beginning. Its applications abound.
Young students can use the triangle to find and explore patterns. High school
33
“Applicaions of Pascal’s Triangle.” Learn Something New Every Day. 4 Oct. 2008
<http://mihirknows.blogspot.com/2007/10/applications-of-pascals-triangle.html>
34
Ohio Department of Education, comp. Academic Content Standards: K-12 Mathematics.
(Columbus, OH: State of Ohio Bard of Education, 2001) 185
35
“Binomial Theorem-Topics in Precalculus.” The Math Page: Topics in Precalculus. 4 Oct.
2008 <http://www.themathpage.com/aprecalc/binomial-theorem.htm>
16
students can use the triangle to complete complicated computations and
increase their knowledge of binomial expansions. From elementary to complex,
Pascal’s Triangle can be used to teach and explore many mathematical subjects.
17
Appendix A
Proof:
1
1
!
!
1
! !
1 !
!
!
1 !
! !
1
1
!
! !
1
1 !
!
· !
1 !
! !
1 !
1 ! !
1
1 !
!
1
!
1 !
1 !
· !
1 ! !
!
· !
1 ! !
1
1 ! !
1 !
!
4 1
4 1 ! !
1 !
1
1
! !
18
Appendix B
Pattern Exploration
Directions: Use a copy of Pascal’s Triangle showing the first 9 rows to answer
the following questions about your general exploration:
1). List any pattern that you find. ______________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
_____________________________________________________________
2).
Predict the next row of numbers and write them here.
3).
Explore the pattern in the sums of the numbers for each row 1-9?
Describe that pattern here: _____________________________________
___________________________________________________________
4).
•
Find the following sets of numbers embedded in the triangle. Describe
where you found them.
Natural numbers {1, 2, 3, 4, …} _________________________________
___________________________________________________________
•
Powers of 2 {1, 2, 4, 8, …} _____________________________________
___________________________________________________________
•
Triangular numbers {1, 3, 6, 10, …} ______________________________
___________________________________________________________
•
Fibonacci numbers {1, 1, 2, 3, 5, 8, …} ____________________________
___________________________________________________________
19
Appendix C
Coloring Multiples
Exploration Questions
Pascal's Triangle is very interesting from a number pattern point of view.
1.
Color all of the odd numbers one color and all of the even numbers another
on your copy of Pascal's Triangle. What do you notice about the pattern?
Have you seen it before?
2.
Use another copy of Pascal’s Triangle to explore other patterns. Try at
least three different multiples. Do you see a general pattern? Describe how each
number you try relates to the pattern for that number.
20
Coin Tossing Activity
Appendix D
Introduction: Coin flipping is based on probability. With an fair coin, the
chances of winning or losing are 50% and consequently, coin flipping is used to
decide such momentous events like who kicks off in a football game. We often
used the terms “it’s a coin toss” or “flip a coin” to describe events that are random
with an equal chance of happening. In this activity students will investigate coin
tossing as it relates to scientific data collection.
According to the Pascal triangle, if you toss 6 coins, you can get one of 7
possibilities with the percent probability of each listed:
Possibility
#1
#2
#3
#4
#5
#6
#7
Outcome
0 heads, 6 tails
1 head, 5 tails
2 heads, 4 tails
3 heads, 3 tails
4 heads, 2 tails
5 heads, 1 tail
6 heads, 0 tails
Occurrence (64 tosses)
1
6
15
20
15
6
1
Percent
1.6%
9.4%
23.4%
31.3%
23.4%
9.4%
1.6%
Based on the laws of probability and with fair coins, students should obtain the
above outcomes. But the factor of random error will influence the results just as it
does in scientific data gathering. Even so, it is obvious that it is very unlikely one
would get 6 heads or 6 tails in a row.
Procedure: The class will break into teams of two or three. Each team will have
a box and lid with 6 pennies, and will do 64 trials. They will record their results.
At the end of the 64 trials, complete a table by totaling each outcome. Calculate
the percent you obtained for each possibility.
•
•
•
Make a histogram of the results and compare it with the mathematically
predicted results. Use Excel or a graphing calculator.
How well did your coin flipping agree with Pascal’s triangle?
Now compare results with other teams. How well did other teams do?
List the occurrences of each team on the board. Does random error affect the
expected results? To overcome the random error, scientists like to take lots of
data. When this is done, it is hoped that the biases over a short run will cancel
each other out. To see this, combine each of the students’ groups into one large
group.
•
•
Does this grand sum more closely approach the expected occurrences
predicted in the Pascal triangle?
Can you see why this is so?
21
Coin Flipping Activity Worksheet
Answer these questions based on your team and the class’s results. Give a
reason for your answers.
1. How well did your coin flipping agree with the theoretical results of Pascal’s
Triangle?
2. How well did other teams’ coin flipping agree with the theoretical results of
Pascal’s Triangle? What experimental factor accounts for the results?
3. How well did grand sum of all the teams agree with the theoretical results of
Pascal’s Triangle?
4. Give a reason why you would expect the grand sum to be closer to the
theoretical percentage.
5. Why do scientists like to take large data sets rather than small ones?
6. Are there any disadvantages to taking large data sets?
22
Tossing Multiple Coins
Appendix E
Complete the table by listing the possible outcomes.
Example: If you toss 2 coins, the possible outcomes are {HH, HT, TH, TT}.
1 coin
0 Tails
1 Tail
2 coins
0 Tails
HH
1 Tail
2 Tails
HT
TH
TT
3 coins
0 Tails
1 Tail
2 Tails
3 Tails
4 coins
0 Tails
1 Tail
2 Tails
3 Tails
4 Tails
5 coins
0 Tails
1 Tail
2 Tails
3 Tails
4 Tails
5 Tails
23
Appendix F
Pizza Problem
It's pizza night for the Shattuck family! Alli heads to LaRosa’s to pick up a large
pizza. However, when she gets there she realizes she didn't check with the
family as to toppings. Rather than imposing her own preferences (pepperoni –
no anchovies!), she decides to call home and ask.
The pizza parlor offers 8 different toppings: anchovies, extra cheese, green
peppers, mushrooms, olives, pepperoni, sausage, and onions. How many
different pizzas could Alli order if a pizza could be selected with any combination
of toppings? Use the following to help you organize your lists.
1)
How many different pizzas could Alli order with only one topping? _____
2)
How many different pizzas can she order each with seven toppings?
_____
3)
Are the number of one-topping pizzas and the number of seven-topping
pizzas related? Why or why not? ________________________________
4)
How many different pizzas can Alli order with two toppings? _________
5)
How many different pizzas can she order with six toppings? ___________
6)
Are the number of two-topping pizzas and the number of six-topping
pizzas related? Why or why not? ________________________________
7)
Find these numbers in Pascal's Triangle.
8)
Use Pascal's Triangle to help you find the number of pizzas that Alli could
order with three, four, and five toppings. ___________________________
9)
Now… How many different pizzas could Alli order if a pizza could be
selected with any combination of toppings?_________________________
24
10)
Can you see another way to approach this problem? What if instead of
figuring out how many pizza topping combinations were possible, Alli
stood and answered the following questions?
•
•
•
•
•
•
•
•
Do you want anchovies?
Do you want extra cheese?
Do you want green peppers?
Do you want mushrooms?
Do you want olives?
Do you want pepperoni?
Do you want sausage?
Do you want onions?
How could this information help Alli find all the different ways a pizza could be
ordered? Describe. ________________________________________________
________________________________________________________________
25
Appendix G
∑
Proof:
by induction
Let
1; then
Let
; then
1
0
0
1
1
1
1
1
1
1
1
1; then
Let
1
1
+
1
1
1
1
…
1
…
1
+
1
1
1
0
0
1
1
1
0
1
1
1
0
∑
1
1
1
1
26
Binomial Coefficients
Appendix H
Problem: Consider
1 . Multiply it out by considering all possible ways to
“thread through” the factors.
•
How many summands are there before you can add like terms? _____
•
How many summands will be of the form “
”? _______
•
How many summands will be of the form “
”? _______
•
How many summands will be of the form “
”? _______
Express the polynomial in standard form: _______________________________
•
Determine the coefficients where ∑
_____
•
______
_______
_______
______
How do these coefficients relate to Pascal’s Triangle? ____________
________________________________________________________
Little Binomial Theorem: Let
Binomial Theorem:
Let
be a whole number. Then,
be a whole number. Then,
1
________
____________
Use the Binomial Theorem to expand each of the following.
1.
5
_______________________________________________________________
2. 1
________________________________________________________________
3.
________________________________________________________________
4.
3
_____________________________________________________________
27
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