The document discusses calculating binomial coefficients without using Pascal's triangle. It begins by stating the formula for calculating binomial coefficients as (n over k)(k factorial) over (n factorial) and proves it using mathematical induction. It then explains how the binomial theorem can be derived from this formula for calculating coefficients. It concludes by providing an example calculation of a binomial coefficient.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
8. Calculating Coefficients
without Pascal’s Triangle
1
0
CLHS n
RHSLHS
Proof
1
!
!
!!0
!
n
n
n
n
RHS
!!
!
knk
n
Ck
n
Step 1: Prove true for k = 0
Hence the result is true for k = 0
15. 0integeraniswherefortrueisresulttheAssume:2Step rrk
!!
!
..
rnr
n
Cei r
n
1fortrueisresulttheProve:3Step rk
!1!1
!
.. 1
rnr
n
Cei r
n
Proof
11
1
r
n
r
n
r
n
CCC
r
n
r
n
r
n
CCC
1
1
1
16. 0integeraniswherefortrueisresulttheAssume:2Step rrk
!!
!
..
rnr
n
Cei r
n
1fortrueisresulttheProve:3Step rk
!1!1
!
.. 1
rnr
n
Cei r
n
Proof
11
1
r
n
r
n
r
n
CCC
r
n
r
n
r
n
CCC
1
1
1
!!
!
!!1
!1
rnr
n
rnr
n
17. 0integeraniswherefortrueisresulttheAssume:2Step rrk
!!
!
..
rnr
n
Cei r
n
1fortrueisresulttheProve:3Step rk
!1!1
!
.. 1
rnr
n
Cei r
n
Proof
11
1
r
n
r
n
r
n
CCC
r
n
r
n
r
n
CCC
1
1
1
!!
!
!!1
!1
rnr
n
rnr
n
!!1
1!!1
rnr
rnn
18. 0integeraniswherefortrueisresulttheAssume:2Step rrk
!!
!
..
rnr
n
Cei r
n
1fortrueisresulttheProve:3Step rk
!1!1
!
.. 1
rnr
n
Cei r
n
Proof
11
1
r
n
r
n
r
n
CCC
r
n
r
n
r
n
CCC
1
1
1
!!
!
!!1
!1
rnr
n
rnr
n
!!1
1!!1
rnr
rnn
!!1
11!
rnr
rnn
22.
!!1
11!
rnr
rnn
!!1
!
rnr
rnn
!1!1
!
rnr
n
Hence the result is true for k = r + 1 if it is also true for k = r
23.
!!1
11!
rnr
rnn
!!1
!
rnr
rnn
!1!1
!
rnr
n
Hence the result is true for k = r + 1 if it is also true for k = r
Step 4: Hence the result is true for all positive integral values of n
by induction
26. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
27. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
28. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
29. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
30. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
31. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
!7!4
!11
4
11
C
32. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
!7!4
!11
4
11
C
1234
891011
33. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
!7!4
!11
4
11
C
1234
891011
34. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
!7!4
!11
4
11
C
1234
891011
3
35. The Binomial Theorem
n
n
nk
k
nnnnn
xCxCxCxCCx 2
2101
n
k
k
k
n
xC
0
integerpositiveaisand
!!
!
where n
knk
n
Ck
n
NOTE: there are (n + 1) terms
This extends to;
n
k
kkn
k
nn
baCba
0
4
11
Evaluate.. Cge
!7!4
!11
4
11
C
1234
891011
330
3
38. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
39. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
40. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
41. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
42. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
43. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
44. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
45. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
4
7
4
11
5
3
5
b
aCT
46. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
4
7
4
11
5
3
5
b
aCT
4
747
4
11
35
b
aC
47. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
4
7
4
11
5
3
5
b
aCT
4
747
4
11
35
b
aC
unsimplified
48. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
4
7
4
11
5
3
5
b
aCT
4
747
4
11
35
b
aC
unsimplified
4
7
8178125330
b
a
49. (ii) Find the value of n so that;
85a) CC nn
kn
n
k
n
CC
13
58
n
n
8
20
87b) CCC nn
k
n
k
n
k
n
CCC
1
11
19n
11
3
5ofexpansionin the5th termtheFind
b
aiii
k
k
kk
b
aCT
3
5
1111
1
4
7
4
11
5
3
5
b
aCT
4
747
4
11
35
b
aC
unsimplified
4
7
8178125330
b
a
4
7
2088281250
b
a