BINOMIAL THEOREM :
AKSHAY MISHRA
XI A , K V 2 , GWALIOR

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example: The coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle, and can be determined by a simple formula involving factorials. These numbers also arise in combinatorics, where the coefficient of xn−kyk is equal to the number of different combinations of k elements that can be chosen from an n-element set.

HISTORY :
HISTORY This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him. The 4th century B.C. Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C. Indian mathematician Pingala to higher orders. A more general binomial theorem and the so-called "Pascal's triangle" were known in the 10th-century A.D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results. Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

STATEMENT OF THE THEOREM :
STATEMENT OF THE THEOREM According to the theorem, it is possible to expand any power of x + y into a sum of the form where denotes the corresponding binomial coefficient. Using summation notation, the formula above can be written This formula is sometimes referred to as the binomial formula or the binomial identity. A variant of the binomial...

...CHAP 1 - Binomial Expansions (Kembangan Binomial)
The binomialtheorem describes the algebraic expansion of powers of a binomial.
Figure 1 : Example use the binomial Expansion in geometric
There are 3 methods to expand binomial expression
Method 1 - Algebra method
Expansion two or more expression.
Example: The expansion depend on power value (n)
n = 0, (a + x)0 = 1
n = 1, (a + x)1 = a + x
n = 2, (a + x)2 = (a + x) (a + x) = a2 + 2ax + x2
n = 3, (a + x)3 = (a + x) (a + x) (a + x) = a3 + 3a2x + 3ax2 + x3
n = 4, (a + x)4 = (a + x)(a + x)(a + x)(a + x) = a4 +4a3x +6a2x2 +4ax3+ x4
Method 2 - PASCAL Triangle
Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician Blaise Pascal
Base on algebra method.
only using the coefficients of terms.
Power value Coefficient
n = 0 1
n = 1 1 1
n = 2 1 2 1
n = 3 1 3 3 1
n = 4 1 4 6 4 1
n = 5 1 5 10 10 5 1
n = 6 1 ? 1
Example:
(1 + 2x)5
n = 5 1...

...The binomialtheorem is a simplified way of finding the expansion of a binomial to a certain power. We can of course find the expanded form of any binomial to a certain power by writing it and doing each step, but this process can be very time consuming when you get into let’s say a binomial to the 10th power.
Example:
(x+y)^0=1 of course because anything to the power if 0 equal 1
(x+y)^1= x+y anything to a power of 1 is just itself.
(x+y)^2= (x+y)(x+y) NOT x^2+y^2.
So expand (x+y)(x+y)=x^2+xy+yx+y^2 or x^2+2xy+y^2.
(x+y)^3=(x+y)(x+y)(x+y) now expanding it is getting quite long. Of course we could do this using the distribution property, but there must be an easier way of expanding binomials with out doing all the steps that is takes to expand something like (x+y)^10.
As surprising as it might be, there is an easier way of find what a binomial equals to a larger power. Using combinations we can find the coefficients of each term. Lets look at an example. The (x+y)^3 was the one I didn’t finish. Let’s look at it now. Using the equation in combination, we can insert the power that we are using and for each term to find the coefficient of each term.
Ex:
This process can be even easier. Blaise Pascal, a famous French mathematical among other things put together a triangle made up of numbers where each number represents the coefficient of each term (when expanded)...

...BINOMIALTHEOREM
OBJECTIVES
Recognize patterns in binomial expansions.
Evaluate a binomial coefficient.
Expand a binomial raised to a power.
Find a particular term in a binomial expansion
Understand the principle of mathematical induction.
Prove statements using mathematical induction.
Definition: BINOMIALTHEOREM
Patterns in Binomial Expansions
A number of patterns, as follows, begin to appear when we write
the binomial expansion of a b n, where n is a positive integer.
a b a b
a b 2 a 2 2ab b 2
a b 3 a3 3a 2b 3ab 2 b3
a b 4 a 4 4a3b 6a 2b2 4ab3 b 4
5
a b a5 5a 4b 10a3b 2 10a 2b3 5ab4 b5
1
and so on.
In each expanded form above, the following can be observed:
n
1. The first term is a , and the exponent on a decreases by 1
in each successive term.
2. The last term is b n and the exponent on b decreases by 1
in each successive term.
3. The sum of the exponents on the variables in any term is
equal to n.
n
n 1 terms in the expanded form of a b .
4. There are
Definition:
Binomial Coefficients
An interesting pattern for the coefficients in the binomial expansion
can be written in the following triangular arrangement
n=0
n=1
n=2
n=3
n=4
n=5
n=6
a bn
1
1
1
1
1...

...1 10/10/01
Fermat’s Little Theorem From the Multinomial Theorem
Thomas J. Osler (osler@rowan.edu) Rowan University, Glassboro, NJ 08028 Fermat’s Little Theorem [1] states that n p −1 − 1 is divisible by p whenever p is prime and n is an integer not divisible by p. This theorem is used in many of the simpler tests for primality. The so-called multinomial theorem (described in [2]) gives the expansion of a multinomial to an integer power p > 0, (a1 + a2 + ⋅⋅⋅ + an ) p = p k1 k2 kn a1 a2 ⋅⋅⋅ an . k1 , k2 , ⋅⋅⋅, kn k1 + k2 +⋅⋅⋅+ kn = p
∑
(1)
Here the multinomial coefficient is calculated by p p! . = k1 , k2 , ⋅⋅⋅, kn k1 !k2 !⋅⋅⋅ kn ! (2)
This is a generalization of the familiar binomialtheorem to the case where the sum of n terms ( a1 + a2 + + an ) is raised to the power p. In (1), the sum is taken over all , kn such that k1 + k2 + + kn = p .
nonnegative integers k1 , k2 ,
In this capsule, we show that Fermat’s Little Theorem can be derived easily from the multinomial theorem. The following steps provide the derivation. 1. All the multinomial coefficients (2) are positive integers. This is clear from the way in which they arise by repeated multiplication by (a1 + a2 + ⋅⋅⋅ + an ) in (1). 2. There are n values of the multinomial coefficient that equal 1. These occur when all but one of the indices kr = 0 , so...

...Four-D? That is, what is the probability of Four-D rejecting a shipment of drives from DataStor?
The probability of Four-D rejecting a shipment of drives from DataStor, if there process is “in control”, is 3.8%.
Four-D conducts the PDQ test on 10 random samples of each shipment.. We used this as the sample size in the Binomial Probability Distribution feature of PhStat.
The probability of the event, that a drive would fall below Four-D’s quality standard of 6.2, was gained from Question 1.
Outcomes 1 – 10 were queried because there are 10 possible scenarios of Four-D rejecting the sample. The cumulative probability of Four-D rejecting a shipment from DataStor’s “in control” process is 3.8%. See the table below for the calculation.
The binomial probability feature was used because this is a decision with two conditions: acceptable or unacceptable.
Binomial Probabilities | | |
| | | |
Data | | |
Sample size | 10 | | |
Probability of an event of interest | 0.0038304 | | |
| | | |
Statistics | | |
Mean | 0.0383038 | | |
Variance | 0.0381571 | | |
Standard deviation | 0.1953384 | | |
| | | |
Binomial Probabilities Table | | |
| X | P(X) | |
| 1 | 0.037 | |
| 2 | 0.00064 | |
| 3 | 6.6E-06 | |
| 4 | 4.4E-08 | |
| 5 | 2E-10 | |
| 6 | 6.5E-13 | |
| 7 | 1.4E-15 | |
| 8 | 2.1E-18 | |
| 9 | 1.8E-21 | |
| 10 |...

...#1 True or false: Even if the sample size is more than 1000, we cannot always use the normal approximation to binomial.
Solution:
If a sample is n>30, we can say that sample size is sufficiently large to assume normal approximation to binomial curve.
Hence the statement is false.
#2
A salesperson goes door-to-door in a residential area to demonstrate the use of a new Household appliance to potential customers. She has found from her years of experience that after demonstration, the probability of purchase (long run average) is 0.30. To perform satisfactory on the job, the salesperson needs at least four orders this week. If she performs 15 demonstrations this week, what is the probability of her being satisfactory? What is the probability of between 4 and 8 (inclusive) orders?
Solution
p=0.30
q=0.70
n=15
k=4
[pic]
Using Megastat we get
| | |15 |
| |0.3 | P |
| | | |
| | |Cumulative |
|k |p(k) |Probability |
|0 |0.00056 |0.00056 |
|1 |0.00503 |0.00559 |
|2 |0.02154 |0.02713 |
|3 |0.05848 |0.08561 |
|4 |0.11278 |0.19838 |
|5 |0.16433 |0.36271 |
|6 |0.18781 |0.55052...

...Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to the Pythagorean Theorem, the sum of the areas of the two red...