This document discusses binomial coefficients and their applications. It provides examples of using binomial coefficients to expand polynomial expressions like (x+y)n and calculate coefficients. Pascal's identity and triangle are also covered, with proofs that the binomial coefficients satisfy C(n+1,k)=C(n,k-1)+C(n,k).
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
Zero. Probabilystic Foundation of Theoretyical PhysicsGunn Quznetsov
No need models - the fundamental theoretical physics is a part of classical probability theory (the part that considers the probability of dot events in the 3 + 1 space-time).
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
THE BINOMIAL THEOREM shows how to calculate a power of a binomial –
(x+ y)n -- without actually multiplying out.
For example, if we actually multiplied out the 4th power of (x + y) --
(x + y)4 = (x + y) (x + y) (x + y) (x + y)
-- then on collecting like terms we would find:
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 . . . . . (1)
Zero. Probabilystic Foundation of Theoretyical PhysicsGunn Quznetsov
No need models - the fundamental theoretical physics is a part of classical probability theory (the part that considers the probability of dot events in the 3 + 1 space-time).
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Necessary and Sufficient Conditions for Oscillations of Neutral Delay Differe...inventionjournals
In this paper, we discuss the oscillatory behavior of all solutions of the first order neutral delay difference equations with several positive and negative coefficients ( ) ( ) ( ) ( ) 0 , i i j j k k I J K x n p x n r x n q x n , o n n (*) where I, J and K are initial segments of natural numbers, pi , rj , qk are positive numbers, i , j are positive integers and k is a nonnegative integer for iI, jJ and kK. We establish a necessary and sufficient conditions for the oscillation of all solutions of (*) is that its characteristic equation ( 1) 1 0 j i k i j k I J K p r q has no positive roots . AMS Subject Classifications : 39A10, 39A12.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Necessary and Sufficient Conditions for Oscillations of Neutral Delay Differe...inventionjournals
In this paper, we discuss the oscillatory behavior of all solutions of the first order neutral delay difference equations with several positive and negative coefficients ( ) ( ) ( ) ( ) 0 , i i j j k k I J K x n p x n r x n q x n , o n n (*) where I, J and K are initial segments of natural numbers, pi , rj , qk are positive numbers, i , j are positive integers and k is a nonnegative integer for iI, jJ and kK. We establish a necessary and sufficient conditions for the oscillation of all solutions of (*) is that its characteristic equation ( 1) 1 0 j i k i j k I J K p r q has no positive roots . AMS Subject Classifications : 39A10, 39A12.
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h.pdfpetercoiffeur18
a) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 1st derivative at nx. Start with a polynomial fit to
points at n-2x , n-1x, nx , n+1x and n+2x .
b) Use Newton’s Polynomials for Evenly Spaced data to derive the O(h4) accurate Second
Centered Difference approximation of the 2nd derivative at nx . Remember, to keep the same
O(h4) accuracy, while taking one more derivative than in Part a, we need to add a point to the
polynomial we used in part a.t,s01530456075y,km0356488107120
Solution
An interpolation assignment generally entails a given set of information points: in which the
values yi can,
xi x0 x1 ... xn
f(xi) y0 y1 ... yn
for instance, be the result of a few bodily measurement or they can come from a long
numerical calculation. hence we know the fee of the underlying characteristic f(x) at the set
of points xi, and we want to discover an analytic expression for f .
In interpolation, the assignment is to estimate f(x) for arbitrary x that lies among the smallest
and the most important xi
. If x is out of doors the variety of the xi’s, then the task is called extrapolation,
which is substantially greater unsafe.
with the aid of far the maximum not unusual useful paperwork utilized in interpolation are the
polynomials.
different picks encompass, as an instance, trigonometric functions and spline features
(mentioned
later during this direction).
Examples of different sorts of interpolation responsibilities include:
1. Having the set of n + 1 information factors xi
, yi, we want to understand the fee of y in the
complete c program languageperiod x = [x0, xn]; i.e. we need to find a simple formulation
which reproduces
the given points exactly.
2. If the set of statistics factors contain errors (e.g. if they are measured values), then we
ask for a components that represents the records, and if feasible, filters out the errors.
3. A feature f may be given within the shape of a pc system which is high priced
to assess. In this case, we want to find a characteristic g which offers a very good
approximation of f and is simpler to assess.
2 Polynomial interpolation
2.1 Interpolating polynomial
Given a fixed of n + 1 records points xi
, yi, we need to discover a polynomial curve that passes
via all the factors. as a consequence, we search for a non-stop curve which takes at the values yi
for every of the n+1 wonderful xi’s.
A polynomial p for which p(xi) = yi whilst zero i n is stated to interpolate the given set of
records points. The factors xi are known as nodes.
The trivial case is n = zero. right here a steady function p(x) = y0 solves the hassle.
The only case is n = 1. In this situation, the polynomial p is a directly line described via
p(x) =
xx1
x0 x1
y0 +
xx0
x1 x0
y1
= y0 +
y1 y0
x1 x0
(xx0)
here p is used for linear interpolation.
As we will see, the interpolating polynomial may be written in an expansion of paperwork,
among
these are the Newton shape and the Lag.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
2. 2
Binomial Coefficients
It allows us to do a quick expansion of
(x+y)n
Why it’s really important:
It provides a good context to present
proofs
Especially combinatorial proofs
3. 3
Let n and r be non-negative integers with
r ≤ n. Then C(n,r) = C(n,n-r)
Or,
Proof (from last slide set):
!
)
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)!
(
!
)
,
(
r
n
n
r
n
n
r
n
n
C
)!
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r
n
r
n
)!
(
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!
)
,
(
r
n
r
n
r
n
C
Review: corollary 1
from section 4.3
r
n
n
r
n
4. 4
Review: combinatorial proof
A combinatorial proof is a proof that uses
counting arguments to prove a theorem,
rather than some other method such as
algebraic techniques
Essentially, show that both sides of the
proof manage to count the same objects
Usually in the form of an English explanation
with supporting formulae
5. 5
Polynomial expansion
Consider (x+y)3:
Rephrase it as:
When choosing x twice and y once, there are
C(3,2) = C(3,1) = 3 ways to choose where the x
comes from
When choosing x once and y twice, there are
C(3,2) = C(3,1) = 3 ways to choose where the y
comes from
3
2
2
3
3
3
3
)
( y
xy
y
x
x
y
x
3
2
2
2
2
2
2
3
)
)(
)(
( y
xy
xy
xy
y
x
y
x
y
x
x
y
x
y
x
y
x
6. 6
Polynomial expansion
Consider
To obtain the x5 term
Each time you multiple by (x+y), you select the x
Thus, of the 5 choices, you choose x 5 times
C(5,5) = 1
Alternatively, you choose y 0 times
C(5,0) = 1
To obtain the x4y term
Four of the times you multiply by (x+y), you select the x
The other time you select the y
Thus, of the 5 choices, you choose x 4 times
C(5,4) = 5
Alternatively, you choose y 1 time
C(5,1) = 5
To obtain the x3y2 term
C(5,3) = C(5,2) = 10
Etc…
5
4
3
2
2
3
4
5
5
5
10
10
5
)
( y
xy
y
x
y
x
y
x
x
y
x
8. 8
Polynomial expansion:
The binomial theorem
For (x+y)n
The book calls this Theorem 1
n
n
n
n
n
y
x
n
y
x
n
y
x
n
n
y
x
n
n
y
x 0
1
1
1
1
0
0
1
1
)
(
n
j
j
j
n
y
x
j
n
0
n
n
n
n
y
x
n
n
y
x
n
n
y
x
n
y
x
n 0
1
1
1
1
0
1
1
0
9. 9
Examples
What is the coefficient of x12y13 in (x+y)25?
What is the coefficient of x12y13 in (2x-3y)25?
Rephrase it as (2x+(-3y))25
The coefficient occurs when j=13:
300
,
200
,
5
!
12
!
13
!
25
12
25
13
25
25
0
25
25
)
3
(
)
2
(
25
)
3
(
2
j
j
j
y
x
j
y
x
00
,545,702,4
33,959,763
)
3
(
2
!
12
!
13
!
25
)
3
(
2
13
25 13
12
13
12
12. 12
Pascal’s Identity
By Pascal’s identity: or 21=15+6
Let n and k be positive integers with n ≥ k.
Then
or C(n+1,k) = C(n,k-1) + C(n,k)
The book calls this Theorem 2
We will prove this via two ways:
Combinatorial proof
Using the formula for
k
n
k
n
k
n
1
1
5
6
4
6
5
7
k
n
13. 13
Combinatorial proof of Pascal’s
identity
Prove C(n+1,k) = C(n,k-1) + C(n,k)
Consider a set T of n+1 elements
We want to choose a subset of k elements
We will count the number of subsets of k elements via 2 methods
Method 1: There are C(n+1,k) ways to choose such a subset
Method 2: Let a be an element of set T
Two cases
a is in such a subset
There are C(n,k-1) ways to choose such a subset
a is not in such a subset
There are C(n,k) ways to choose such a subset
Thus, there are C(n,k-1) + C(n,k) ways to choose a subset of k
elements
Therefore, C(n+1,k) = C(n,k-1) + C(n,k)
14. 14
Rosen, section 4.4, question 19:
algebraic proof of Pascal’s identity
k
n
k
n
k
n
1
1
)!
)(
1
(
)
1
(
!
)
1
(
)!
1
(
)!
(
*
)
1
(
)!
1
(
k
n
k
n
k
n
n
n
n
k
n
k
n
k
n
1
1
n
n
)!
(
!
!
))!
1
(
(
)!
1
(
!
)!
1
(
!
)!
1
(
k
n
k
n
k
n
k
n
k
n
k
n
)!
(
)!
1
(
!
)!
)(
1
(
)!
1
(
!
)!
)(
1
(
)!
1
(
!
)
1
(
k
n
k
k
n
k
n
k
n
k
n
k
n
k
n
k
k
n
n
k
k
n
k
n
k
n 1
)
1
(
1
)
1
(
)
1
(
1
1
k
n
k
n
)
1
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)
1
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)
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1
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k
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k
k
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k
n
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k
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k
n
Substitutions:
15. 15
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usefulness for our users, Google Gulp will send packets
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17. 17
Proof practice: corollary 1
Let n be a non-negative integer. Then
Algebraic proof
n
k
n
k
n
0
2
n
k
k
n
k
k
n
0
1
1
n
n
)
1
1
(
2
n
k k
n
0
n
j
j
j
n
n
y
x
j
n
y
x
0
)
(
18. 18
Proof practice: corollary 1
Let n be a non-negative integer. Then
Combinatorial proof
A set with n elements has 2n subsets
By definition of power set
Each subset has either 0 or 1 or 2 or … or n elements
There are subsets with 0 elements, subsets with 1
element, … and subsets with n elements
Thus, the total number of subsets is
Thus,
n
k
n
k
n
0
2
n
n
k k
n
2
0
0
n
n
k k
n
0
1
n
n
n
20. 21
Proof practice: corollary 2
Let n be a positive integer. Then
Algebraic proof
This implies that
n
0
0
n
k
k
k
n
0
0
)
1
(
5
3
1
4
2
0
n
n
n
n
n
n
n
k
k
n
k
k
n
0
1
)
1
(
n
1
)
1
(
k
n
k k
n
)
1
(
0
21. 22
Proof practice: corollary 3
Let n be a non-negative integer. Then
Algebraic proof
n
k
k
k
n
k
n
0
2
1
n
k
n
k
k
n
0
3
2
n
n
)
2
1
(
3
n
k
k
k
n
0
2
22. 23
Vandermonde’s identity
Let m, n, and r be non-negative integers
with r not exceeding either m or n. Then
The book calls this Theorem 3
r
k k
n
k
r
m
r
n
m
0
23. 24
Combinatorial proof of
Vandermonde’s identity
Consider two sets, one with m items and one with n
items
Then there are ways to choose r items from the union of
those two sets
Next, we’ll find that value via a different means
Pick k elements from the set with n elements
Pick the remaining r-k elements from the set with m elements
Via the product rule, there are ways to do that for EACH
value of k
Lastly, consider this for all values of k:
Thus,
r
k k
n
k
r
m
r
n
m
0
r
n
m
k
n
k
r
m
r
k k
n
k
r
m
0
24. 25
Review of Rosen, section
4.3, question 11 (a)
How many bit strings of length 10 contain
exactly four 1’s?
Find the positions of the four 1’s
The order of those positions does not matter
Positions 2, 3, 5, 7 is the same as positions 7, 5, 3, 2
Thus, the answer is C(10,4) = 210
Generalization of this result:
There are C(n,r) possibilities of bit strings of length n
containing r ones
25. 26
Yet another combinatorial proof
Let n and r be non-negative integers with r ≤ n.
Then
The book calls this Theorem 4
We will do the combinatorial proof by showing
that both sides show the ways to count bit
strings of length n+1 with r+1 ones
From previous slide: achieves this
n
r
j r
j
r
n
1
1
1
1
r
n
26. 27
Yet another combinatorial proof
Next, show the right side counts the same objects
The final one must occur at position r+1 or r+2 or … or
n+1
Assume that it occurs at the kth bit, where r+1 ≤ k ≤ n+1
Thus, there must be r ones in the first k-1 positions
Thus, there are such strings of length k-1
As k can be any value from r+1 to n+1, the total number
of possibilities is
Thus,
r
k 1
1
1
1
n
r
k r
k
n
r
k r
k
n
r
j r
j
n
r
j r
j
r
n
1
1
27. 28
Rosen, section 4.4, question 24
Show that if p is a prime and k is an integer such that
1 ≤ k ≤ p-1, then p divides
We know that
p divides the numerator (p!) once only
Because p is prime, it does not have any factors less than p
We need to show that it does NOT divide the
denominator
Otherwise the p factor would cancel out
Since k < p (it was given that k ≤ p-1), p cannot divide k!
Since k ≥ 1, we know that p-k < p, and thus p cannot
divide (p-k)!
Thus, p divides the numerator but not the denominator
Thus, p divides
k
p
)!
(
!
!
k
p
k
p
k
p
k
p
28. 29
Rosen, section 4.4, question 38
Give a combinatorial proof that if n is positive
integer then
Provided hint: show that both sides count the
ways to select a subset of a set of n elements
together with two not necessarily distinct
elements from the subset
Following the other provided hint, we express
the right side as follows:
2
0
2
2
)
1
(
n
n
k
n
n
k
n
k
1
2
0
2
2
2
)
1
(
n
n
n
k
n
n
n
k
n
k
29. 30
Rosen, section 4.4, question 38
Show the left side properly counts the
desired property
n
k k
n
k
0
2
Choosing a subset of k
elements from a set of
n elements
Consider each
of the possible
subset sizes k
Choosing one of
the k elements in
the subset twice
30. 31
Rosen, section 4.4, question 38
Two cases to show the right side: n(n-1)2n-2+n2n-1
Pick the same element from the subset
Pick that one element from the set of n elements: total of n possibilities
Pick the rest of the subset
As there are n-1 elements left, there are a total of 2n-1 possibilities to pick a given
subset
We have to do both
Thus, by the product rule, the total possibilities is the product of the two
Thus, the total possibilities is n*2n-1
Pick different elements from the subset
Pick the first element from the set of n elements: total of n possibilities
Pick the next element from the set of n-1 elements: total of n-1 possibilities
Pick the rest of the subset
As there are n-2 elements left, there are a total of 2n-2 possibilities to pick a given
subset
We have to do all three
Thus, by the product rule, the total possibilities is the product of the three
Thus, the total possibilities is n*(n-1)*2n-2
We do one or the other
Thus, via the sum rule, the total possibilities is the sum of the two
Or n*2n-1+n*(n-1)*2n-2
31. 32
Quick survey
I felt I understood the material in this slide set…
a) Very well
b) With some review, I’ll be good
c) Not really
d) Not at all
32. 33
Quick survey
The pace of the lecture for this slide set was…
a) Fast
b) About right
c) A little slow
d) Too slow
33. 34
Quick survey
How interesting was the material in this slide
set? Be honest!
a) Wow! That was SOOOOOO cool!
b) Somewhat interesting
c) Rather borting
d) Zzzzzzzzzzz