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 +...

...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...

...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...

...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...

...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...

...#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...

...The 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...