The binomial theorem provides a formula for expanding binomial expressions of the form (a + b)^n. It states that the terms of the expansion are determined by binomial coefficients. Pascal's triangle is a mathematical arrangement that shows the binomial coefficients and can be used to determine the coefficients in a binomial expansion. The proof of the binomial theorem uses mathematical induction to show that the formula holds true for any positive integer value of n.
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)
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
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)
A short presentation to explain the use of permutations and combinations and some examples to illustrate the concepts. This was made as an assignment in which i was to explain the concepts to the class.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
SET
A set is a well defined collection of objects, called the “elements” or “members” of the set.
A specific set can be defined in two ways-
If there are only a few elements, they can be listed individually, by writing them between curly braces ‘{ }’ and placing commas in between. E.g.- {1, 2, 3, 4, 5}
The second way of writing set is to use a property that defines elements of the set.
e.g.- {x | x is odd and 0 < x < 100}
If x is an element o set A, it can be written as ‘x A’
If x is not an element of A, it can be written as ‘x A’
Special types of sets-
Standard notations used to define some sets:
N- set of all natural numbers
Z- set of all integers
Q- set of all rational numbers
R- set of all real numbers
C- set of all complex numbers
TYPES OF SETS
-subset
-singleton set
-universal set
-empty set
-finite set
-infinte set
-eual set
-disjoint set
-cardinal set
-power set
OPERATIONS ON SET
The four basic operations are:
1. Union of Sets
2. Intersection of sets
3. Complement of the Set
4. Cartesian Product of sets
Union of two given sets is the smallest set which contains all the elements of both the sets.
A B = {x | x A or x B}
Let a and b are sets, the intersection of two sets A and B, denoted by A B is the set consisting of elements which are in A as well as in B
A B = {X | x A and x B}
If A B= , the sets are said to be disjoint.
If U is a universal set containing set A, then U-A is called complement of a set.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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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.
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.
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.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Create Map Views in the Odoo 17 ERPCeline George
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2. Binomials
An expression in the form a + b is called a binomial,
because it is made of of two unlike terms.
We could use the FOIL method repeatedly to evaluate
expressions like (a + b)2, (a + b)3, or (a + b)4.
– (a + b)2 = a2 + 2ab + b2
– (a + b)3 = a3 + 3a2b + 3ab2 + b3
– (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
But to evaluate to higher powers of (a + b)n would be a
difficult and tedious process.
For a binomial expansion of (a + b)n, look at the
expansions below:
– (a + b)2 = a2 + 2ab + b2
– (a + b)3 = a3 + 3a2b + 3ab2 + b3
– (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
• Some simple patterns emerge by looking at these
examples:
– There are n + 1 terms, the first one is an and the last is bn.
– The exponent of a decreases by 1 for each term and the
exponents of b increase by 1.
– The sum of the exponents in each term is n.
3. For bigger exponents
To evaluate (a + b)8, we will find a way to calculate the
value of each coefficient.
(a + b)8= a8 + __a7b + __a6b2 + __a5b3 + __a4b4 + __a3b5 + __a2b6 + __ab7 + b8
– Pascal’s Triangle will allow us to figure out what the coefficients
of each term will be.
– The basic premise of Pascal’s Triangle is that every entry (other
than a 1) is the sum of the two entries diagonally above it.
The Factorial
In any of the examples we had done already, notice that
the coefficient of an and bn were each 1.
– Also, notice that the coefficient of an-1 and a were each n.
These values can be calculated by using factorials.
– n factorial is written as n! and calculated by multiplying the
positive whole numbers less than or equal to n.
Formula: For n≥1, n! = n • (n-1) • (n-2)• . . . • 3 • 2 • 1.
Example: 4! = 4 3 2 1 = 24
– Special cases: 0! = 1 and 1! = 1, to avoid division by zero in the
next formula.
4. The Binomial Coefficient
To find the coefficient of any term of (a +
b)n, we can apply factorials, using the
formula:
n
n!
n Cr
r
r! n r !
Blaise Pascal
(1623-1662)
– where n is the power of the binomial
expansion, (a + b)n, and
– r is the exponent of b for the specific term we are
calculating.
So, for the second term of (a + b)8, we would have n = 8
and r = 1 (because the second term is ___a7b).
– This procedure could be repeated for any term we choose, or all of
the terms, one after another.
– However, there is an easier way to calculate these coefficients.
Example :
7 C3
7!
7!
7
(7 3)! • 3! 4! • 3! 4! • 3!
(7 • 6 • 5 • 4) • (3 • 2 • 1)
(4 • 3 • 2 • 1) • (3 • 2 • 1)
7•6•5• 4
4 • 3 • 2 •1
35
5. Recall that a binomial has two terms...
(x + y)
The Binomial Theorem gives us a quick method to expand
binomials raised to powers such as…
(x + y)0
(x + y)1
(x + y)2
(x + y)3
Study the following…
Row
Row
Row
Row
Row
Row
Row
0
1
This triangle is called Pascal’s
1
Triangle (named after mathematician
1 1
Blaise Pascal).
2
1 2 1
3
1 3 3 1
Notice that row 5 comes from adding up
4
1 4 6 4 1 row 4’s adjacent numbers.
(The first row is named row 0).
5
1 5 10 10 5 1
6 1 6 15 20 15 6 1
This pattern will help us find the coefficients when we expand binomials...
6. Finding coefficient
What we will notice is that when r=0 and when r=n, then
nCr=1, no matter how big n becomes. This is because:
n C0
n!
n 0 ! 0!
n!
1
n! 0!
n Cn
n!
n n ! n!
n!
1
0! n!
Note also that when r = 1 and r = (n-1):
n
C1
n!
n 1 ! 1!
n n 1!
n 1 ! 1!
n
n Cn
1
n
n!
n 1 ! n 1!
n n 1!
1! n 1 !
So, the coefficients of the first and last terms will always be
one.
– The second coefficient and next-to-last coefficient will be n.
(because the denominators of their formulas are equal)
n
7. Constructing Pascal’s Triangle
Continue evaluating nCr for n=2 and n=3.
When we include all the possible values of r such that
0≤r≤n, we get the figure below:
n=0
0C0
n=1
1C0 1C1
n=2
2C0
n=3
n=4
3C0
4C0
n=5
5C0
n=6
6C0 6C1
3C1
4C1
5C1
2C1
6C2
3C2
4C2
5C2
2C2
4C3
5C3
6C3
3C3
4C4
5C4
6C4
5C5
6C5
6C6
8. Knowing what we know about nCr and its values when
r=0, 1, (n-1), and n, we can fill out the outside values
of the Triangle:
r=n, nCr=1
r=1, nCr=n
r=(n-1), nCr=n
n=0
1
0C0
n=1
r=0, nCr=1
1 0 1C
1C1 1C1 1
1
n=2
1 1 2 2C 11 C
1
2C01 C2 1 2C2 2
2
n=3
1 0 3C33 33C 111C
C2 3
3
31
3C111 C1 3C2 2 3C3 3
n=4
1 0 4CC 44C 44C 111C
C3 4
4
4 14 C2
4C111 4 1 4C2 2 4C3 3 4C4 4
n=5
1 0 5C55 55C 55C 55C 111C
54 5
51
2
3
5C111 C1 5C2 2 5C3 3 5C4 4 5C5 5
n=6
1 0 6CC 66C 66C 66C 66C 111C
C3 C4 C5 6
6
6 16 C2
6C111 6 1 6C2 2 6C3 3 6C4 4 6C5 5 6C6 6
9. Using Pascal’s Triangle
We can also use Pascal’s Triangle to expand
binomials, such as (x - 3)4.
The numbers in Pascal’s Triangle can be used to find
the coefficients in a binomial expansion.
For example, the coefficients in (x - 3)4 are represented
by the row of Pascal’s Triangle for n = 4.
x
3
4
4 C0 x
1x
4
4
1
3
0
4 x
4 C1 x
3
3
3
4
6
4
1
3
6 x
1
2
4 C2 x
9
2
4 x
3
1
2
1
4 C3 x
27
1x 4 12x 3 54x 2 108x 81
1x
1
0
3
81
3
4 C4 x
0
3
4
10. The Binomial Theorem
( x y)n
with nCr
x n nx n 1 y nCr x n r y r nxy n 1 y n
n!
(n r )!r !
The general idea of the Binomial Theorem is that:
– The term that contains ar in the expansion (a + b)n is
n
n
r n r
r
ab
or
n!
arbn
n r ! r!
r
– It helps to remember that the sum of the exponents of each term
of the expansion is n. (In our formula, note that r + (n - r) = n.)
Example: Use the Binomial Theorem to expand (x4 + 2)3.
(x 4
2)3
4 3
C0(x )
3
4 3
1 (x )
4 2
C1( x ) (2)
3
4
2
C2(x )( 2)
3
3 ( x 4 ) 2 (2) 3 (x 4 )( 2) 2
x12 6 x8 12 x 4 8
1 (2)
(2)
3 C3
3
3
11. Example:
Find the eighth term in the expansion of (x + y)13 .
Think of the first term of the expansion as x13y 0 .
The power of y is 1 less than the number of the term in
the expansion.
The eighth term is 13C7 x 6 y7.
13
C7
13!
6! • 7!
(13 • 12 • 11 • 10 • 9 • 8) • 7!
6! • 7!
13 • 12 • 11 • 10 • 9 • 8
1716
6 • 5 • 4 • 3 • 2 •1
Therefore,
the eighth term of (x + y)13 is 1716 x 6 y7.
12. Proof of Binomial Theorem
Binomial theorem for any positive integer n,
a b
n
n
c0an
n
c1a n 1b nc2an 2b2 ........ ncnbn
Proof
The proof is obtained by applying principle of mathematical
induction.
Step: 1
Let the given statement be
f (n) : a b
n
n
c0an
n
c1an 1b nc2an 2b2 ........ ncnbn
Check the result for n = 1 we have
f (1) : a b
1
1
c0a1 1c1a1 1b1 a b
Thus Result is true for n =1
Step: 2
Let us assume that result is true for n = k
f (k ) : a b
k
k
c0ak
k
c1ak 1b k c2ak 2b2 ........ k ck bk
13. Step: 3
We shall prove that f (k + 1) is also true,
k 1
f (k 1) : a b
k 1
c0ak
1
k 1
c1ak b
k 1
c2ak 1b2 ........ k 1ck 1bk
Now,
a b
k 1
(a b)( a b) k
k
a b
c0 a k
k
c1a k 1b k c2 a k 2b 2 ........
k
ck b k
From Step 2
k
c0 a k
1
1
k
c1a k b k c2 a k 1b 2 ........ k ck ab k
k
k
c0 a k
c0 a k b k c1a k 1b 2 ........ k ck 1ab k
k
c1
k
c0 a k b
k
c2
...
by using
k 1
c0
1, k cr
k
cr
k
1
k
k
ck b k
1
c1 a k 1b 2 .....
k
ck
k
ck 1 ab k
cr , and k ck
1
k
ck b k
1
k 1
ck
1
1
14. k 1
c0 a k
1
k 1
c1a k b
k 1
c2 a k 1b 2 ........
k 1
ck ab k
k 1
ck 1b k
Thus it has been proved that f(k+1) is true when ever
f(k) is true,
Therefore, by Principle of mathematical induction f(n) is
true for every Positive integer n.
1
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