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Topic: Binomial expansion

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	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-09-19)
		Usually, binomial expansion is introduced using a construction called Pascal's Triangle, but I prefer to think of it in terms of something called the binomial coefficient, which I'll explain later.
		Is it a coincidence that the (n)th row corresponds to the coefficients of the expansion (x+y)^n?
		In essence, the most common use of binomial expansion is in expression simplification; the expansion is usually done when you expect terms to cancel.
	mathforum.org /dr.math/problems/fama.7.10.96.html (871 words)

	the Binomial theorem
		The Pascal triangle is a table, which happens to be in the shape of a triangle, of the coefficient of the binomial expansion of (x + y)^n, where n is a natural number.
		Theorem: The coefficients of the binomial expansion of (x + y)^n, indeed, are given by the rows of the Pascal triangle.
		Theorem: The coefficients of the binomial expansion of (x + y)^n, indeed, are the C(n, m).
	www.rism.com /Trig/binomial.htm (789 words)

	Binomial Distribution
		Each term in the binomial equation is the probability of the corresponding possible outcome.
		Binomial problems can be solved using the Binomial table and basic probability principles.
		Use the binomial table in your text to find the probability of exactly 8 heads in 10 tosses of a fair coin.
	www.louisville.edu /~aslajo01/Week13.htm (369 words)

	Binomial theorem - Topics in precalculus
		The upper index n is the exponent of the expansion; the lower index k indicates which term.
		We found the binomial coefficients to be 1 5 10 10 5 1.
		In the binomial, x is "a", and −1 is "b".
	www.themathpage.com /aPreCalc/binomial-theorem.htm (873 words)

	Binomial Expansion - Uncyclopedia
		Binomial Expansion is actually the corruption of the word BiGnomial Expansion, which was an exciting sport played up until the 1636 Winter Sports Reforms Act.
		There have been several attempts to bring back BiGnomial Expansion, the most recent being in the 1960s, but these 'hippies' spoke in a cryptic language involving sunbeams and flowers.
		Unfortunately they were taken as lunatics and expelled from society, but they now live underground and have taken the form of Damon from the Wizard of Oz.
	uncyclopedia.org /wiki/Binomial_Expansion (319 words)

	MCDB 2150 -- Lecture 5
		Binomial probability: In many cases, we are interested only in the probability of obtaining a particular final result, such as two heads in three tosses of a coin, and not in the order of the individual events that yield the final result.
		Thus, binomial expansions are often expressed in terms of (p + q) raised to the power n.
		Binomial expansion: The key feature of a binomial expansion is that it always deals with exactly two mutually exclusive events whose combined probabilities must equal exactly 1.0.
	www.colorado.edu /MCDB/MCDB2150Fall/notes/L05.html (2101 words)

	SparkNotes: Binomial Expansion: Introduction and Summary
		This chapter deals with binomial expansion; that is, with writing expressions of the form (a + b)
		It is used in statistics to calculate the binomial distribution.
		Binomial expansion is also interesting from a mathematical point of view--it gives mathematicians insight into the properties of polynomials.
	www.sparknotes.com /math/algebra2/binomialexpansion/summary.html (253 words)

	Binomial theorem - Wikipedia, the free encyclopedia
		This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century.
		It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayyám in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.
		The binomial theorem is mentioned in the Gilbert and Sullivan song "I am the Very Model of a Modern Major General".
	en.wikipedia.org /wiki/Binomial_expansion (542 words)

	Binomial Distribution (Site not responding. Last check: 2007-09-19)
		The Binomial Distribution describes the probability distribution for an event happening or not happening over the course of N trials.
		In the general case, there are N trials in which an event has a probability of p occurring and a probability of q of not occurring.
		We can calculate the probablity for obtaining each number of heads by expanding the binomial equation (p a + q b)^N. The coeifficients in the resulting expansion correspond to the probabilities for 0..N heads or positive events.
	retina.anatomy.upenn.edu /~lance/modelmath/binomial.html (295 words)

	Exponents
		Any power of a binomial can be obtained from the Binomial Theorem.
		This is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression.
		The binomial expansion is a useful example of a series.
	hyperphysics.phy-astr.gsu.edu /hbase/alg3.html (105 words)

	Algebra 2
		(2) each subsequent coefficient is gotten by multiplying the exponent on the first term of the binomial by the coefficient and dividing by the number of the term.
		(3) the coefficient on the first term of the binomial increases by 1 on each term and the coefficient of the second term of the binomial decreases by 1.
		Note: If the binomial has a negative sign between the terms, simply alternate signs in the expansion.
	webpages.charter.net /jdsanders/sandersmeth.htm (290 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		binomial -------- Binomial.TXT Version 1.00 3 December 1998 ============ This program runs on the TI-89 and computes terms of the binomial expansion of (a + x)^n.
		Running the program =================== Execute binomial() from the command line, which brings up an identifying screen in which you are asked to press Enter to continue.
		If "Float 6" is not suitable for your particular purposes you may edit the program easily, although you should not run it in "Fix anything" as this disrupts the indirection used for saving the terms and totals.
	learning.mgccc.cc.ms.us /math/89/binomial.txt (806 words)

	PlanetMath: binomial theorem
		See Also: binomial formula, binomial coefficient, binomial distribution
		This is version 12 of binomial theorem, born on 2001-10-16, modified 2005-02-22.
		It might be worth pointing out that this theorem also holds if a and b belong to an commutative rig (the spelling is right; I really mean "rig", not "ring" here) since we only use some basic algebraic properties of real or complex numbers in the proof.
	planetmath.org /encyclopedia/BinomialTheorem.html (117 words)

	MCDB 2150 -- Lecture 22
		Thus, binomial expansions are often expressed in terms of (p + q) raised to the power n, as is done in our current textbook.
		In order to keep the equations arising out of the binomial expansion as simple as possible, the symbol y is frequently used to replace n-x as the number of occurrences of the alternative event.
		Binomial expansion is needed for complex calculations: For calculations that only involve a few branch points, the forked line approach provides a convenient way to visualize the relationships among the individual events that are involvled.
	www.colorado.edu /MCDB/MCDB2150Fall/notes00/L0022.html (2472 words)

	Binomial Theorem -- from Wolfram MathWorld
		There are several closely related results that are variously known as the binomial theorem depending on the source.
		Pascal's pamphlet, together with his correspondence on the subject with Fermat beginning in 1654 (and published in 1679) is the basis for naming the arithmetical triangle in his honor.
		Whittaker, E. and Robinson, G. "The Binomial Theorem." §10 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed.
	mathworld.wolfram.com /BinomialTheorem.html (399 words)

	Binomial Expansion
		A binomial is an expression that contains two terms.
		For example, in the last line, the third number is 10, which is the sum of 4 and 6.
		There is a way of obtaining the binomial coefficients directly, and this is by using the binomial theorem.
	library.thinkquest.org /C0110248/algebra/biexpintro.htm (180 words)

	The Binomial Theorem
		The Binomial Theorem is an important theorem, useful across a wide range of mathematics.
		The theorem enables you to calculate all the terms of this expansion in your head.
		the Binomial Theorem generalises to the situation where n is a number other than a positive integer, but this is outside the scope of this course.
	www.maths.abdn.ac.uk /~igc/tch/eg1006/notes/node17.html (314 words)

	binomial
		The binomial theorem gives the result of raising a binomial expression to a power; the expansion and the series it leads to are called the binomial expansion and the binomial series.
		A binomial distribution is described by a formula related to the binomial expansion.
		A binomial equation is a particular kind of equation that contains two terms.
	www.daviddarling.info /encyclopedia/B/binomial.html (137 words)

	The Binomial Expansion
		The sum of the powers of "a" and "b" in each term is n.
		The coefficients of the binomial expansion are called
		The coefficients can be arranged in a triangular array, which is called Pascal's Triangle.
	www.sac.edu /homepages/kashi_majid/Binomial_Expansion.htm (75 words)

	Pascal's Simplices Project
		Section II considers arrays of numbers which reside in higher dimensional space, so it may be helpful to be familiar with simplices as geometric objects.
		These binomial expansions can be completely specified by their coefficients, since the determination of the exponents is straightforward: In the expansion of (a + b)
		The first property to notice about Pascal's triangle is that we can compute its entries without actually working out binomial expansions, for each number in Pascal's triangle is the sum of the two numbers that lie above it to the left and right.
	www.math.rutgers.edu /~erowland/pascalssimplices-project.html (994 words)

	Leaving Cert. Higher Level Maths - Sequences And Series - Pascal's Triangle And The Binomial Theorem
		Pascal's Triangle And The Binomial Theorem - By Conor Mc Dermottroe
		where n is the power of the binomial and r is the number of the term.
		Calculating the full binomial expansion is just a matter of generating each term in the expansion, one at a time.
	www.netsoc.tcd.ie /~jgilbert/maths_site/applets/sequences_and_series/pascal_s_triangle_and_the_binomial_theorem.html (155 words)

	4. The Binomial Theorem
		A binomial is an algebraic expression containing 2 terms.
		The LiveMath expansion of factorials is usually better than our calculators, in that it can go higher.
		NOTE (1): This is an infinite series, where the binomial theorem deals with a finite expansion.
	www.intmath.com /SereBino/4_BiTh.php (389 words)

	Jay's Corner - 2 (Site not responding. Last check: 2007-09-19)
		Notice that each term of the binomial expansions is a product composed of one item of each color.
		We form one of the terms in the binomial expansion by picking either an x or a y from each of the boxes and multiplying our selections together.
		Because the numbers C(n,k) appear as the coefficients of the terms in a binomial expansion, they are called binomial coefficents.
	www-math.cudenver.edu /~wcherowi/jcorn3.html (689 words)

	Binomial Expansion
		The binomial expansion is a tool for estimating the probability that a specific combination of mutually exclusive outcomes will occur.
		The mutually exclusive outcomes might, for example, be female versus male; and the specific combination of outcomes might be 3 females out of 6 offspring.
		The general formula for the binomial expansion is:
	www.mhhe.com /biosci/cellmicro/hartwell/binomialexpansion.mhtml (260 words)

	Newton and the binomial theorem
		In 1665, plague was raging in England, and Isaac Newton, a new (and undistinguished) graduate of the University of Cambridge, was forced to spend most of the next two years in the relative safety of his family's country manor in Woolsthorpe.
		In that 18-month period of retreat, he came up with his proof and extension of the binomial theorem, invented calculus (which he called his "method of fluxions"), discovered the law of universal gravitation, and proved that white light is composed of all colors.
		Newton's consideration of infinite series and the notion of limit through the binomial theorem led directly to his development of calculus.
	www.ualr.edu /lasmoller/newton.html (505 words)

	★ Maths Part 2 - Binomial Expansion ★
		For any number of crossings, the terms can be derived by the expansion of Equation 4a.
		Fortunately, the coefficients of the expansion could be expressed in a general term, which is based on its significance: the number of combinations which satisfy this requirement.
		With Equation 4b, the binomial expansion could be re-written in a more general expression.
	www.geocities.com /lynnp27/binomial01.html (75 words)

	7.5 - The Binomial Theorem
		A binomial is a polynomial with two terms.
		Combinations will be discussed more fully in section 7.6, but here is a brief summary to get you going with the Binomial Expansion Theorem.
		The Binomial Expansion Theorem can be written in summation notation, where it is very compact and manageable.
	www.richland.cc.il.us /james/lecture/m116/sequences/binomial.html (550 words)

	Binomial Expansion
		For this expansion, you need the 'combinations' function on the TI83, but if you don't have this (or can't find it), use the (non-user-friendly) formula in the book.It's in the form
		A binomial expansion is in the form (a+x)
		However, when a is not 1, you have to rearrange, so (x-2) becomes -2(1+(x/-2)), and in the expansion, replace all x for (x/-2), and the multiply the whole expnsion by -2.
	www.hewett.norfolk.sch.uk /curric/maths/ALEV_MAT/94cb1204/expansi.htm (112 words)

	[No title]
		Use the formula to find the coefficient of the third term in the expansion of (x - 3)7.
		Find the 6th term in the expansion of (3x + 2)8.
		Use the Binomial Theorem to find the numerical value of (0.998)6 correct to five decimal places.
	www.mc.maricopa.edu /~geraldine/150_extras/9_5BinTh.doc (225 words)

	Pascal's Triangle and the Binomial Expansion ...........................
		Now what we'd like to look at is how to work out a binomial expansion for any exponent.
		One of its uses is to work out binomial expansions like the one above.
		To do a binomial expansion, you need to know that the rows correspond to the power, beginning with 0.
	www.worsleyschool.net /science/files/pascal/triangle.html (754 words)

	The Binomial Distribution
		It is called the Binomial Distribution (I bet you didn't see that coming!) and is used to give the probabilities of various combinations of successes and failures in a certain number of trials.
		Let's consider the second term of that expansion, the one that gives the probability of 4 red beads and a fl bead.
		Of course, they are the numbers in front of the terms of the binomial expansion.
	members.tripod.com /~RichardBowles/maths/binomial/binomial.htm (1686 words)

	7.5 - The Binomial Theorem
		A binomial is a polynomial with two terms.
		The Binomial Expansion Theorem can be written in summation notation, where it is very compact and manageable.
		When you go to use the binomial expansion theorem, it's actually easier to put the guidelines from the top of this page into practice.
	www.richland.edu /james/lecture/m116/sequences/binomial.html (550 words)

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