The document contains solutions to permutation and combination problems. It lists the number of total outcomes for rolling a dice n times as 6n, and the number of total outcomes for tossing a coin (n-1) times as 2n-1. It then works through additional problems calculating permutations, combinations, and compound probabilities.
This will help you on how to solve quadratic equations by factoring.
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This will help you in evaluating summation notation.
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How do you calculate angle between 2 lines in 3 dimensional geometry?
Learn this and much more in this presentation/video. Here, we learn how to calculate angle between 2 lines, equation of a line passing through 2 points and perpendicular to a line and how to find the foot of a perpendicular from a point to a line.
These are useful for grade 12 mathematics students and students appearing for the NATA eligibility test in architecture which has maths as a section.
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Arithmetic to Analytic Geometry!
Before learning CALCULUS there are 10 points you need to reconsider as you continue your journey to the college life.
This exam offers word problems which includes branches like trigonometry, logarithms, functions, algebra, arithmetic and so forth. It ranges from 7th Grade to 10th Grade. It assess your basic knowledge of numbers and analytical skills. Hurry up and try!
GMAT Quant Strategy- What to expect in Quantitative Reasoning Section on the ...CrackVerbal
Learn some practice tips & strategies from our experts on how to crack GMAT Quant! Shortcuts, success formulas & Math mantras- all to take you a step ahead of the GMAT competition!
This will help you on how to solve quadratic equations by factoring.
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https://tinyurl.com/ycjp8r7u
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This will help you in evaluating summation notation.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
How do you calculate angle between 2 lines in 3 dimensional geometry?
Learn this and much more in this presentation/video. Here, we learn how to calculate angle between 2 lines, equation of a line passing through 2 points and perpendicular to a line and how to find the foot of a perpendicular from a point to a line.
These are useful for grade 12 mathematics students and students appearing for the NATA eligibility test in architecture which has maths as a section.
For more videos, subscribe to my channel or visit my page
https://www.mathmadeeasy.co/about
Arithmetic to Analytic Geometry!
Before learning CALCULUS there are 10 points you need to reconsider as you continue your journey to the college life.
This exam offers word problems which includes branches like trigonometry, logarithms, functions, algebra, arithmetic and so forth. It ranges from 7th Grade to 10th Grade. It assess your basic knowledge of numbers and analytical skills. Hurry up and try!
GMAT Quant Strategy- What to expect in Quantitative Reasoning Section on the ...CrackVerbal
Learn some practice tips & strategies from our experts on how to crack GMAT Quant! Shortcuts, success formulas & Math mantras- all to take you a step ahead of the GMAT competition!
this presentation covers the topic percentage, profit and loss aptitude questions in level 1. (basic) categorywise the techniques are supported by suitable examples
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More companies in the process of recruitment, play more emphasis in the topic of numbers in numerical aptitude. Especially for AMCAT aspirants this is very much useful.
Some standard questions asked in cognizant aptitude tests recently has been sorted with answers. it will be beneficial to other company preparation aptitude also.
This presentation uses the technology of Microsoft Multiple Mouse Mischief software. If you need assistance, visit microsoft site for multiple mouse support. Probability aptitude questions level 2
This is a collection of fifty questions from important topics in Aptitude where students should pay more attention and practice. Questions taken from various net sources. Some of the answers were edited. This presentation could be run only in office 2010 or latest.
Problems on Trains is the Aptitude topic which most of the companies prefer to ask. Here students could find some examples on the different categories of problems on trains.
Time & Distance is a broader topic in aptitude. Here the moving object could be train or person or boats etc.,
Students could find useful techniques to solve time & distance aptitude problems.
Higher studies after plustwo for the tamil medium students in tamilnadu. This is the first part of career guidance which was given by me in kaliappa goundenpudur, pollachi.
this document is useful for all students preparing for placement. this document contains aptitude techniques. for each technique examples are shown. since i could not upload the ppt due to some technical problem the arrangment of text will not be in order.
examples are not stereo typed. all the techniques are taken from various internet sources hence could not be acknowledge individually. those who have any objections please mention in the comment box so that those particular portion could be removed.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
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.
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!
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.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
2. PERMUTATION
12 June 2015C.S.VEERARAGAVAN 9894834264 VEERAA1729@GMAIL.COM 2
When a dice is rolled for n times, then the number of total
out comes as
(1) 6 (2) 6n-1(3) 6n (4)6n+1
The number of total out comes is 6n
01
3. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
3
When a coin is tossed for (n-1) times, then the
number of total out comes is
(1) 2 (2) 2n (3) 2n+1 (4)2n-1
the number of total out comes is 2n-1
02
5. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
5
If nCr = nPr then the value of r can be
(1) 0 (2) 1
(3) 2 (4) More than one of the above
nPr = nCr .* r!
Hence r could be zero or one.
07
(4) More than one of the above
7. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
7
How many words can be formed using all the letters of
the word GINGER?
(1) 720 (2) 240 (3) 380 (4)360
Number of letters = 6
G repeated two times.
Hence number of words =
6!
2!
=3 x 4 x 5 x 6 =360
09
8. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
8
How many different words, which begin with N can be
formed using all the letters of the word COUNTRY?
(1) 720 (2) 120 (3) 24 (4)5040
The word COUNTRY has 7 letters.
Since the first letter is N, the remaining 6 places can be
filled with 6 letters in 6! Ways. = 720.
11
9. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
9
How many three-digit numbers can be formed using the digits
{1,2,3,4,5} , so that each digit is repeated any number of times?
(1) 150 (2) 200 (3) 25 (4)125
Consider 3 blanks
Since there are 5 digits, each blank can be filled in 5 ways.
Total number of ways is (5) (5) (5) = 125.
14
10. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
10
All possible four-digit numbers, with distinct digits are
formed, using the digits {1,3,4,5,6}.
How many of them are divisible by 5?
(1) 8 (2) 12 (3) 24 (4)20
Consider four blanks
The units place is filled with 5.
The remaining three blanks can be filled with 4 digits in 4P3
ways.
The number of four digit numbers required is 4 (3) (2) or 24.
16
11. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
11
In how many ways can 3 boys and 2 girls be seated in arrow,
so that all girls sit together?
(1) 12 (2) 24 (3) 84 (4)48
Let All the girls be one unit.
There are 3 boys and 1 unit of girls arranged in 4! Ways.
The two girls can be arranged among themselves in 2! Ways.
Total number of arrangements 4! 2! = 24 (2) = 48.
17
12. PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
12
In how many ways can 4 boys be seated in 6 chairs ?
(1)180 (2) 720 (3) 360 (4)240
4 boys be seated in 6 chairs in 6P4 = 360 ways.
22
13. PERMUTATION WITH RESTRICTIONS
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
13
A basket contains 3 bananas and 4 apples. In how many
ways can one select one or more fruits?
(1) 17 (2)20 (3) 18 (4)19
Bananas can be selected in 4 ways and
Apples can be selected in 5 ways
Number of ways of selecting fruits is 4 x 5 = 20.
Number of ways of selecting one or more = 20 – 1 = 19.
30
14. PERMUTATION WITH RESTRICTIONS
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
14
How many different words can be formed using the letters
of the word MARKET so that they begin with K and end
with R?
(1) 42 (2) 24 (3) 12 (4)64
Since the first place and the last place are to be filled with
K and R, the remaining four places can be arranged
In 4! Or 24 ways.
26
15. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
15
When a coin is tossed for n times then the number
of ways of getting exactly ‘r’ heads is
(1) 2r (2) nCr (3) nPr (4)2n
03
17. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
17
If nC5=nC7, then 2n+1C2 is
(1) 250 (2) 300 (3) 240 (4)280
nCr=nCs r + s = n
5 + 7 = 12
25C2 =
25 X 24
2
= 300
06
18. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
18
The number of ways of selecting 4 members from a
group of 10 members so that one particular member is
always included is
(1) 63 (2) 72 (3) 84 (4)56
Since one particular member is always included, we have
to select 3 members from 9 members.
This can be done in 9C3 = 84 ways.
10
19. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
19
In how many ways can two consonants be selected from
the English alphabet?
(1) 420 (2) 105 (3) 210 (4)300
There are 21 consonants.
Two consonants can be selected from 21 consonants in
21C2 = 210
12
20. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
20
A bag contains 3 white balls, 4 green balls and 5 red balls.
In how many ways two balls can be selected?
(1) 132 (2) 66 (3) 33 (4)76
Total balls = 12.
12C2 = 66 ways.
13
21. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
21
Ten points are selected on a plane, such that no three of
them are collinear. How many different straight lines can
be formed by joining these points?
(1) 54 (2) 45 (3) 90 (4)108
A straight line is formed by joining any two points.
Two points can be selected from 10 points in 10C2 ways.
i.e., 45 ways.
18
22. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
22
If the number of diagonals of a polygon is 5 times the number
of sides, the polygon is
(1) 13 (2) 20 (3) 15 (4)17
Let the number of sides be n.
Number of diagonals =
n(n−3)
2
Given
n(n−3)
2
= 5n,
N-3 = 10
N = 13
19
23. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
23
Rahul has 6 friends. In how many ways can he
invite 5 or more friends for dinner?
(1) 1 (2) 6 (3) 7 (4)8
5 friends can be invited in 6C5 ways.
6 friends can be invited in 6C6 ways.
Total = 6C5 + 6C6 = 6 + 1 = 7
29
24. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
24
In how many ways can four letters be selected from the
word EDUCATION ?
(1)9C8 (2)9C7 (3) 9C4 (4) 9C6
The number of letters in the word EDUCATION is 9.
4 letters can be selected from these 9 letters in 9C4 ways.
24
25. COMBINATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
25
In how many ways can 3 blue balls be selected from a
bag which contains 4 white balls and 6 blue balls ?
(1) 20 (2) 10 (3) 120 (4)210
The number of blue balls is 6.
3 balls can be selected from 6 blue balls in 6C3 ways = 20
27
26. CIRCULAR PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
26
In how many ways can 5 men and 3 women be seated
around a circular table?
(1) 720 (2) 5040 (3) 4020 (4)2520
N persons can be positioned around a circle in (n-1)!
ways.
Hence 8 persons can be arranged in 7! = 5040 ways.
20
27. 12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
27
How many odd numbers can be formed using the digits
{0,2,4,6} ?
(1)0 (2) 192 (3) 18 (4)20
Since all given numbers are even, we could not get any
odd number.
15
28. CIRCULAR PERMUTATION
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
29
If n books can be arranged on an ordinary shelf in 720 ways,
then in how many ways can these books be arranged on a
circular shelf ?
(1)120 (2) 720 (3) 360 (4)60
N books can be arranged in n! ways.
Given n! = 720 = 6!
N = 6
6 books can be arranged on a circular shelf in (6-1)! =5!
120 ways.
23
29. COMPOUND EVENT
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
30
In how many ways can 4 letters be posted into 3 letters
boxes ?
(1)32 (2) 64 (3) 27 (4)81
Each letter can be posted in 3 ways.
Four letters can be posted in 3 x 3 x 3 x 3 or 81 ways.
25
30. COMPOUND EVENT
12 June 2015
C.S.VEERARAGAVAN 9894834264
VEERAA1729@GMAIL.COM
31
When two coins are tossed and a cubical dice is rolled,
then the total outcomes for the compound event is?
(1)42 (2) 24 (3) 28 (4)10
When two coins are tossed there are 4 possible
outcomes.
When a dice is rolled there are 6 possible outcomes.
Hence there are 4 x 6 = 24 outcomes.
28