Binomial theorem
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The binomial coefficients appear as the entries of Pascal’s triangle where each entry is the sum of the two above it.
In elementary algebra , the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial . According to the theorem, it is possible to expand the polynomial (x + y)^{n} into a sum involving terms of the form a x^{b} y^{c}, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example (for n = 4),
The coefficient a in the term of a x^{b} y^{c} is known as the binomial coefficient
$$\displaystyle \tbinom nb
or
$$\displaystyle \tbinom nc
(the two have the same value). These coefficients for varying n and b can be arranged to form Pascal’s triangle . These numbers also arise in combinatorics , where
$$\displaystyle \tbinom nb
gives the number of different combinations of b elements that can be chosen from an nelement set .
Contents
 1 History
 2 Theorem statement
 3 Examples
 3.1 Geometric explanation
 4 Binomial coefficients
 4.1 Formulae
 4.2 Combinatorial interpretation
 5 Proofs
 5.1 Combinatorial proof
 5.1.1 Example
 5.1.2 General case
 5.2 Inductive proof
 5.1 Combinatorial proof
 6 Generalizations
 6.1 Newton’s generalized binomial theorem
 6.2 Further generalizations
 6.3 Multinomial theorem
 6.4 Multibinomial theorem
 6.5 General Leibniz rule
 7 Applications
 7.1 Multipleangle identities
 7.2 Series for e
 7.3 Probability
 8 The binomial theorem in abstract algebra
 9 In popular culture
 10 See also
 11 Notes
 12 References
 13 Further reading
 14 External links
History[ edit ]
Special cases of the binomial theorem were known since at least the 4th century B.C. when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2.^{ [1] }^{ [2] } There is evidence that the binomial theorem for cubes was known by the 6th century in India.^{ [1] }^{ [2] }
Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. 200 B.C.), which contains a method for its solution.^{ [3] }^{:230} The commentator Halayudha from the 10th century A.D. explains this method using what is now known as Pascal’s triangle .^{ [3] } By the 6th century A.D., the Indian mathematicians probably knew how to express this as a quotient
$$\displaystyle \frac n!(nk)!k!
,^{ [4] } and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara .^{ [4] }
The first formulation of the binomial theorem and the table of binomial coefficients, to our knowledge, can be found in a work by AlKaraji , quoted by AlSamaw’al in his “alBahir”.^{ [5] }^{ [6] } AlKaraji described the triangular pattern of the binomial coefficients^{ [7] } and also provided a mathematical proof of both the binomial theorem and Pascal’s triangle, using an early form of mathematical induction .^{ [7] } The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.^{ [2] } The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui ^{ [8] } and also Chu ShihChieh .^{ [2] } Yang Hui attributes the method to a much earlier 11th century text of Jia Xian , although those writings are now also lost.^{ [3] }^{:142}
In 1544, Michael Stifel introduced the term “binomial coefficient” and showed how to use them to express
$$\displaystyle (1+a)^n
in terms of
$$\displaystyle (1+a)^n1
, via “Pascal’s triangle”.^{ [9] } Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique (1653). However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia , and Simon Stevin .^{ [9] }
Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.^{ [9] }^{ [10] }
Theorem statement[ edit ]
According to the theorem, it is possible
to expand any power of x + y into a sum of the form
where each
$$\displaystyle \tbinom nk
is a specific positive integer known as a binomial coefficient . (When an exponent is zero, the corresponding power expression is taken to be 1 and this multiplicative factor is often omitted from the term. Hence one often sees the right side written as
$$\displaystyle \binom n0x^n+\ldots
.) This formula is also referred to as the binomial formula or the binomial identity. Using summation notation , it can be written as
The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical.
A simple variant of the binomial formula is obtained by substituting 1 for y, so that it involves only a single variable . In this form, the formula reads
or equivalently
Examples[ edit ]
Pascal’s triangle
The most basic example of the binomial theorem is the formula for the square of x + y:
The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal’s triangle. (Note that the top “1” of the triangle is considered to be row 0, by convention.) The coefficients of higher powers of x + y correspond to lower rows of the triangle:
Several patterns can be observed from these examples. In general, for the expansion (x + y)^{n}:
 the powers of x start at n and decrease by 1 in each term until they reach 0 (with x^{0} = 1, often unwritten);
 the powers of y start at 0 and increase by 1 until they reach n;
 the nth row of Pascal’s Triangle will be the coefficients of the expanded binomial when the terms are arranged in this way;
 the number of terms in the expansion before like terms are combined is the sum of the coefficients and is equal to 2^{n}; and
 there will be n + 1 terms in the expression after combining like terms in the expansion.
The binomial theorem can be applied to the powers of any binomial. For example,
For a binomial involving subtraction, the theorem can be applied by using the form (x − y)^{n} = (x + (−y))^{n}. This has the effect of changing the sign of every other term in the expansion:
Geometric explanation[ edit ]
Visualisation of binomial expansion up to the 4th power
For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes.
In calculus , this picture also gives a geometric proof of the derivative
$$\displaystyle (x^n)’=nx^n1:
^{ [11] } if one sets
$$\displaystyle a=x
and
$$\displaystyle b=\Delta x,
interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an ndimensional hypercube ,
$$\displaystyle (x+\Delta x)^n,
where the coefficient of the linear term (in
$$\displaystyle \Delta x
) is
$$\displaystyle nx^n1,
the area of the n faces, each of dimension
$$\displaystyle (n1):
Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms,
$$\displaystyle (\Delta x)^2
and higher, become negligible, and yields the formula
$$\displaystyle (x^n)’=nx^n1,
interpreted as
 “the infinitesimal rate of change in volume of an ncube as side length varies is the area of n of its
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus , one obtains Cavalieri’s quadrature formula , the integral
$$\displaystyle \textstyle \int x^n1\,dx=\tfrac 1nx^n
– see proof of Cavalieri’s quadrature formula for details.^{ [11] }
Binomial coefficients[ edit ]
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written
$$\displaystyle \tbinom nk
, and pronounced “n choose k”.
Formulae[ edit ]
The coefficient of x^{n−k}y^{k} is given by the formula
which is defined in terms of the factorial function n!. Equivalently, this formula can be written
with k factors in both the numerator and denominator of the fraction . Note that, although this formula involves a fraction, the binomial coefficient
$$\displaystyle \tbinom nk
is actually an integer .
Combinatorial interpretation[ edit ]
The binomial coefficient
$$\displaystyle \tbinom nk
can be interpreted as the number of ways to choose k elements from an nelement set. This is related to binomials for the following reason: if we write (x + y)^{n} as a product
then, according to the distributive law , there will be one term in the expansion for each choice of either x or y from each of the binomials of the product. For example, there will only be one term x^{n}, corresponding to choosing x from each binomial. However, there will be several terms of the form x^{n−2}y^{2}, one for each way of choosing exactly two binomials to contribute a y. Therefore, after combining like terms , the coefficient of x^{n−2}y^{2} will be equal to the number of ways to choose exactly 2 elements from an nelement set.
Proofs[ edit ]
Combinatorial proof[ edit ]
Example[ edit ]
The coefficient of xy^{2} in
equals
$$\displaystyle \tbinom 32=3
because there are three x,y strings of length 3 with exactly two y’s, namely,
corresponding to the three 2element subsets of 1, 2, 3 , namely,
where each subset specifies the positions of the y in a corresponding string.
General case[ edit ]
Expanding (x + y)^{n} yields the sum of the 2^{ n} products of the form e_{1}e_{2} … e_{ n} where each e_{ i} is x or y. Rearranging factors shows that each product equals x^{n−k}y^{k} for some k between 0 and n. For a given k, the following are proved equal in succession:
 the number of copies of x^{n − k}y^{k} in the expansion
 the number of ncharacter x,y strings having y in exactly k positions
 the number of kelement subsets of 1, 2, …, n
This proves the binomial theorem.
Inductive proof[ edit ]
Induction yields another proof of the binomial theorem. When n = 0, both sides equal 1, since x^{0} = 1 and
$$\displaystyle \tbinom 00=1
.
Now suppose that the equality holds for a given n; we will prove it for n + 1.
For j, k ≥ 0, let [ƒ(x, y)]_{ j,k} denote the coefficient of x^{j}y^{k} in the polynomial ƒ(x, y).
By the inductive hypothesis, (x + y)^{n} is a polynomial in x and y such that [(x + y)^{n}]_{ j,k} is
\displaystyle \tbinom nk
if j + k = n, and 0 otherwise.
The identity
shows that (x + y)^{n + 1} also is a polynomial in x and y, and
since if j + k = n + 1, then (j − 1) + k = n and j + (k − 1) = n. Now, the right hand side is
by Pascal’s identity .^{ [12] } On the other hand, if j +k ≠ n + 1, then (j – 1) + k ≠ n and j +(k – 1) ≠ n, so we get 0 + 0 = 0. Thus
which is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step.
Generalizations[ edit ]
Newton’s generalized binomial theorem[ edit ]
Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series . In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number r, one can define
where
$$\displaystyle (\cdot )_k
is the Pochhammer symbol , here standing for a falling factorial . This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with x > y,^{ [Note 1] } and r is any complex number, one has
When r is a nonnegative integer, the binomial coefficients for k > r are zero, so this equation reduces to the usual binomial theorem, and there are at most r + 1 nonzero terms. For other values of r, the series typically has infinitely many nonzero terms.
For example, r = 1/2 gives the following series for the square root:
Taking
$$\displaystyle r=1
, the generalized binomial series gives the geometric series formula , valid for
$$x
:
More generally, with r = −s:
So, for instance, when
$$\displaystyle s=1/2
,
Further generalizations[ edit ]
The generalized binomial theorem can be extended to the case where x and y are complex numbers. For this version, one should again assume x > y^{ [Note 1] } and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius x centered at x.
The generalized binomial theorem is valid also for elements x and y of a Banach algebra as long as xy = yx, x is invertible, and y/x < 1.
A version of the binomial theorem is valid for the following Pochhammer symbol like family of polynomials: for a given real constant c, define
$$\displaystyle x^(0)=1
and
$$\displaystyle x^(n)=\prod _k=1^n[x+(k1)c]
for
$$\displaystyle n>0
. Then^{ [13] }
The case c = 0 recovers the usual binomial theorem.
More generally, a sequence
$$\displaystyle \p_n\_n=0^\infty
of polynomials is said to be binomial if
An operator
$$\displaystyle Q
on the space of polynomials is said to be the basis operator of the sequence
$$\displaystyle \p_n\_n=0^\infty
if
$$\displaystyle Qp_0=0
and
$$\displaystyle Qp_n=np_n1
for all
$$\displaystyle n\geqslant 1
. A sequence
$$\displaystyle \p_n\_n=0^\infty
is binomial if and only if its basis operator is a Delta operator .^{ [14] } Writing
$$\displaystyle E^a
for the shift by
$$\displaystyle a
operator, the Delta operators corresponding to the above “Pochhammer” families of polynomials are the backward difference
$$\displaystyle IE^c
for
$$\displaystyle c>0
, the ordinary derivative for
$$\displaystyle c=0
, and the forward difference
$$\displaystyle E^cI
for
$$\displaystyle c<0
.
Multinomial theorem[ edit ]
The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is
where the summation is taken over all sequences of nonnegative integer indices k_{1} through k_{m} such that the sum of all k_{i} is n. (For each term in the expansion, the exponents must add up to n). The coefficients
$$\displaystyle \tbinom nk_1,\cdots ,k_m
are known as multinomial coefficients, and can be computed by the formula
Combinatorially, the multinomial coefficient
$$\displaystyle \tbinom nk_1,\cdots ,k_m
counts the number of different ways to partition an nelement set into disjoint subsets of sizes k_{1}, …, k_{m}.
Multibinomial theorem[ edit ]
It is often useful when working in more dimensions, to deal with products of binomial expressions. By the binomial theorem this is equal to
This may be written more concisely, by multiindex notation , as
General Leibniz rule[ edit ]
The general Leibniz rule gives the nth derivative of a product of two functions in a form similar to that of the binomial theorem:^{ [15] }
Here, the superscript (n) indicates the nth derivative of a function. If one sets f(x) = e^{ax} and g(x) = e^{bx}, and then cancels the common factor of e^{(a + b)x} from both sides of the result, the ordinary binomial theorem is recovered.
Applications[ edit ]
Multipleangle identities[ edit ]
For the complex numbers the binomial theorem can be combined with De Moivre’s formula to yield multipleangle formulas for the sine and cosine . According to De Moivre’s formula,
Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos(nx) and sin(nx). For example, since
De Moivre’s formula tells us that
which are the usual doubleangle identities. Similarly, since
De Moivre’s formula yields
In general,
and
Series for e[ edit ]
The number e is often defined by the formula
Applying the binomial theorem to this expression yields the usual infinite series for e. In particular:
The kth term of this sum is
As n → ∞, the rational expression on the right approaches one, and therefore
This indicates that e can be written as a series:
Indeed, since each term of the binomial expansion is an increasing function of n, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to e.
Probability[ edit ]
The binomial theorem is closely related to the probability mass function of the negative binomial distribution . The probability of a (countable) collection of independent Bernoulli trials
$$\displaystyle \X_t\_t\in S
with probability of success
$$\displaystyle p\in [0,1]
all not happening is
A useful upper bound for this quantity is
$$\displaystyle e^pn
. ^{ [16] }
The binomial theorem in abstract algebra[ edit ]
Formula (1) is valid more generally for any elements x and y of a semiring satisfying xy = yx. The theorem is true even more generally: alternativity suffices in place of associativity .
The binomial theorem can be stated by saying that the polynomial sequence 1, x, x^{2}, x^{3}, … is of binomial type .
In popular culture[ edit ]
 The binomial theorem is mentioned in the MajorGeneral’s Song in the comic opera The Pirates of Penzance .
 Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem .
 The Portuguese poet Fernando Pessoa , using the heteronym Álvaro de Campos , wrote that “Newton’s Binomial is as beautiful as the Venus de Milo . The truth is that few people notice it.”^{ [17] }
 In the 2014 film The Imitation Game , Alan Turing makes reference to Isaac Newton’s work on the Binomial Theorem during his first meeting with Commander Denniston at Bletchley Park.
See also[ edit ]
 Binomial approximation
 Binomial distribution
 Binomial inverse theorem
 Stirling’s approximation
Notes[ edit ]
 ^ ^{a} ^{b} This is to guarantee convergence. Depending on r, the series may also converge sometimes when x = y.
References[ edit ]
 ^ ^{a} ^{b} Weisstein, Eric W. “Binomial Theorem” . Wolfram MathWorld.
 ^ ^{a} ^{b} ^{c} ^{d} Coolidge, J. L. (1949). “The Story of the Binomial Theorem”. The American Mathematical Monthly. 56 (3): 147–157. doi : 10.2307/2305028 . JSTOR 2305028 .
 ^ ^{a} ^{b} ^{c} JeanClaude Martzloff; S.S. Wilson; J. Gernet; J. Dhombres (1987). A history of Chinese mathematics. Springer.
 ^ ^{a} ^{b} Biggs, N. L. (1979). “The roots of combinatorics”. Historia Math. 6 (2): 109–136. doi : 10.1016/03150860(79)900740 .
 ^ “Taming the unknown. A history of algebra from antiquity to the early ttwentieth century” (PDF). Bulletin of the American Mathematical Society: 727.
However, algebra advanced in other respects. Around 1000, alKaraji stated the binomial theorem
 ^ Rashed, R. (19940630). The Development of Arabic Mathematics: Between Arithmetic and Algebra . Springer Science & Business Media. p. 63. ISBN 9780792325659 .
 ^ ^{a} ^{b} O’Connor, John J. ; Robertson, Edmund F. , “Abu Bekr ibn Muhammad ibn alHusayn AlKaraji” , MacTutor History of Mathematics archive , University of St Andrews .
 ^ Landau, James A. (19990508). “Historia Matematica Mailing List Archive: Re: [HM] Pascal’s Triangle” (mailing list email). Archives of Historia Matematica. Retrieved 20070413.
 ^ ^{a} ^{b} ^{c} Kline, Morris (1972). History of mathematical thought. Oxford University Press. p. 273.
 ^ Bourbaki, N. (18 November 1998). Elements of the History of Mathematics Paperback. J. Meldrum (Translator). ISBN 9783540647676 .
 ^ ^{a} ^{b} Barth, Nils R. (2004). “Computing Cavalieri’s Quadrature Formula by a Symmetry of the nCube”. The American Mathematical Monthly. 111 (9): 811–813. doi : 10.2307/4145193 . ISSN 00029890 . JSTOR 4145193 , author’s copy , further remarks and resources
 ^ Binomial theorem – inductive proofs Archived February 24, 2015, at the Wayback Machine .
 ^ Sokolowsky, Dan; Rennie, Basil C. (February 1979). “Problem 352” (PDF). Crux Mathematicorum. 5 (2): 55–56.
 ^ Aigner, Martin (1997) [Reprint of the 1979 Edition]. Combinatorial Theory. Springer. p. 105. ISBN 3540617876 .
 ^ Seely, Robert T. (1973). Calculus of One and Several Variables. Glenview: Scott, Foresman. ISBN 9780673077790 .
 ^ Cover, Thomas M.; Thomas, Joy A. (20010101). Data Compression. John Wiley & Sons, Inc. p. 320. doi : 10.1002/0471200611.ch5 . ISBN 9780471200611 .
 ^ “Arquivo Pessoa: Obra Édita – O binómio de Newton é tão belo como a Vénus de Milo” . arquivopessoa.net.
Further reading[ edit ]
 Bag, Amulya Kumar (1966). “Binomial theorem in ancient India”. Indian J. History Sci. 1 (1): 68–74.
 Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). “(5) Binomial Coefficients”. Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153–256. ISBN 9780201558029 . OCLC 17649857 .
External links[ edit ]
The Wikibook Combinatorics has a page on the topic of: The Binomial Theorem 
 Solomentsev, E.D. (2001) [1994], “Newton binomial” , in Hazewinkel, Michiel , Encyclopedia of Mathematics , Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Binomial Theorem by Stephen Wolfram , and “Binomial Theorem (StepbyStep)” by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project , 2007.
This article incorporates material from inductive proof of binomial theorem on PlanetMath , which is licensed under the Creative Commons Attribution/ShareAlike License .
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The Sections: The formulas,
The Binomial Theorem is The formal expression of
Yeah, I know; that formula Recall that the factorial
As you might imagine, drawing I could never remember (3x � + _{10}C_{3} + _{10}C_{6} + _{10}C_{9} Note how the highlighted Your first step, given (1)(59049)x^{10}(1) + (210)(729)x^{6}(16) + (120)(27)x^{3}(�128) = 59049x^{10} + 1088640x^{4} As painful as the BinomialTheorem Top



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Video: How to Use the Binomial Theorem to Expand a Binomial
In this video lesson, you will see what the binomial theorem has in common with Pascal’s triangle. Learn how you can use Pascal’s triangle to help you to easily expand a binomial.
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The Binomial Theorem
In math, we usually have formulas we can work from. Sometimes, these formulas can be somewhat complicated. But many times, these formulas have patterns to them. If we can spot the patterns, we can find an easier way to use the formula. This is what we are going to do in this video. We will learn the patterns in the binomial theorem and how we can use it to expand our binomials that are multiplied with themselves numerous times.
Our binomial theorem is a theorem that tells us what happens when you multiply a binomial by itself numerous times. Recall that a binomial is a polynomial made up of two terms. To keep things simple for us, though, we will label our binomial as (a + b) where a and b are your two terms in the binomial.
Think of the number of wheels a bicycle has and you will be able to remember the number of terms a binomial has. The formula that is connected with the Binomial Theorem is this one. We have a plus b to the nth power is equal to the summation of n choose k times a to the n minus k power times b to the kth power from k equals 0 to k equals n.
Yes, I know, it looks complicated. But it does produce patterns that you can easily remember.
Understanding the Formula
Let’s dissect this formula so that we can understand it better. The first part, a plus b to the nth power, we understand to be the type of problem we can use our formula with. For example, we can use our formula with problems such as (x + y)^3 and (a + b)^5 because they follow the form of the first part of the formula.
The next part of the formula, the part after the equal sign, tells us our answer for these types of problems. The big symbol in front is called the summation symbol. It tells you to sum up the part of the formula that is to the right of it starting from k = 0 and going until k = n. We will usually see a k and/or an n in the formula. For each k = 0, 1, 2, etc., until n, we will plug in our values and then add up all the terms together to find our final answer. I will show you an example shortly so that you can better understand it.
The part of the formula that is inside the parentheses is notation for a factorial problem. We read it as n choose k. It is short for n!/(k!(n – k)!). So, if n = 4 and k = 2, this part would equal 4 times 3 times 2 times 1 in the numerator and 2 factorial times 4 minus 2 factorial, which is 2 factorial times 2 factorial in the denominator. So, the denominator is 2 times 1 times 2 times 1. Multiplying all these out, we find that it equals 6.
Now, let’s see an example of how this binomial theorem formula works. We will follow the formula to expand (a + b)^3.
Looks long doesn’t it? If we do our subtractions and our factorial calculations, we end up with this:
Hey, our final answer doesn’t look so bad. Can we spot any patterns here? Yes, we can. Look at the exponents of our a and our b. Notice that as we go to the right, the exponent of our a starts at our n and decreases until we get to 0. The exponent of our b, on the other hand, starts at 0 and increases until it gets to n.
In our case, our n is 3, so our a exponent started with 3 and went down to 0, and our b exponent started at 0 and went up to 3. Remember that when the exponent is 0, it equals 1. So b^0 equals 1. And when it equals 1, we won’t write it since there are other things in the term that we write. So, what other patterns can we spot?
Pascal’s Triangle
This is where Pascal’s triangle comes in. Pascal’s triangle is a triangle of numbers where the first and last terms of a row are 1 and all the other numbers are the sum of the two numbers directly above it. Pascal’s Triangle always begins with 1 at the tip and two 1’s underneath, forming a triangle. The next row has a 1 at both ends. The middle number is the sum of the two numbers above it, so 1 + 1 equals 2. The next row will also have 1’s at either end.
The numbers in between these 1’s are made up of the sum of the two numbers above it. So, the first number is 1 + 2, which is 3. The next number is 1 + 2, which is also 3. So, this row’s numbers are 1, 3, 3, and 1. We can keep going to make our Pascal’s Triangle even bigger by continuing our addition.
Notice the numbers in the row with the 1, 3, 3, and 1. Doesn’t our answer have those same numbers as coefficients? Yes, it does. As it turns out, the coefficients from the binomial theorem follow the numbers of Pascal’s triangle. Each row would give the coefficients a different n. The first row, the tip, is for n = 0.
The next one is n = 1. The one after that is n = 2. And the next is n = 3. And we can keep going that way. So, if we want to expand the binomial (a + b)^5, our coefficients according to Pascal’s Triangle are 1, 5, 10, 10, 5, and 1, in that order.
Expanding a Binomial
Now that we know all the patterns involved, we can expand our binomials much easier. What are the patterns, again? They are that the exponents of the a term decrease to 0 starting from n, while the exponents of b increase starting from 0 and ending with n. Our coefficients follow a particular row of Pascal’s triangle.
Let’s see about using these patterns to expand the binomial (a + b)^5. Our n equals 5, so we will use the coefficients from the row of Pascal’s triangle that corresponds to n = 5. This particular row has the numbers 1, 5, 10, 10, 5, and 1. So this means that we will have a total of 6 terms in our answer. The other patterns that we will use are the ones for the exponents.
We know that for our a the exponent begins at n, 5 in our case, and goes down until it is 0, while our b, the exponent begins at 0 and goes up until it is 5 or n. So, using all these patterns, we find that our answer equals a to the fifth power plus 5 times a to the fourth power times b plus 10 times a to the third power times b to the second power plus 10 times a to the second power times b to the third power plus 5 times a times b to the fourth power plus b to the fifth power.
This is our answer. See how it follows all the patterns that we talked about? Not bad, eh?
Lesson Summary
Let’s review what we’ve learned now. We’ve learned that the binomial theorem is a theorem that tells us what happens when you multiply a binomial by itself numerous times. The formula is a plus b to the nth power is equal to the summation of n choose k times a to the n minus k power times b to the kth power from k equals 0 to k equals n. While the formula looks kind of scary, we see that it gives us some very useful patterns that we can use to help us.
We see that the exponents of the first term of our binomial begins at our n and goes down until it is 0, while the exponents of our second term begins at 0 and goes up until it is n. We also see that our coefficients follow a particular row of Pascal’s triangle, which is a triangle of numbers where the first and last terms of a row are 1 and all the other numbers are the sum of the two numbers directly above it. Using these patterns, we can easily expand any binomial.
Learning Outcomes
Learning the topics in this lesson could enable you to:
 Recall the binomial theorem
 Recognize and use Pascal’s triangle to expand a binomial
 Expand a binomial using the binomial theorem
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Introduction to Sequences: Finite and Infinite
4:57
How to Use Factorial Notation: Process and Examples
4:40
How to Use Series and Summation Notation: Process and Examples
4:16
Arithmetic Sequences: Definition & Finding the Common Difference
5:55
How and Why to Use the General Term of an Arithmetic Sequence
5:01
The Sum of the First n Terms of an Arithmetic Sequence
6:00
Understanding Arithmetic Series in Algebra
6:17
Working with Geometric Sequences
5:26
How and Why to Use the General Term of a Geometric Sequence
5:14
The Sum of the First n Terms of a Geometric Sequence
4:57
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4:41
Using Recursive Rules for Arithmetic, Algebraic & Geometric Sequences
5:52
Using Sigma Notation for the Sum of a Series
4:44
Mathematical Induction: Uses & Proofs
7:48
How to Find the Value of an Annuity
4:49
How to Use the Binomial Theorem to Expand a Binomial
8:43
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