What is Binomial theorem: Definition and 137 Discussions

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 axbyc, 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),




(
x
+
y

)

4


=

x

4


+
4

x

3


y
+
6

x

2



y

2


+
4
x

y

3


+

y

4


.


{\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}
The coefficient a in the term of axbyc is known as the binomial coefficient







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
or







(


n
c


)






{\displaystyle {\tbinom {n}{c}}}
(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







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore







(


n
b


)






{\displaystyle {\tbinom {n}{b}}}
is often pronounced as "n choose b".

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  1. RChristenk

    Find the ##r^{th}## term of ##(a+2x)^n##

    ##r^{th}## term from beginning: ##^nC_{r-1}a^{n-r+1}(2x)^{r-1}## For the ##r^{th}## term from the end, we first know there are a total of ##n+1## terms in this binomial expansion. Subtracting the (##r^{th}## term from the end) from the total number of terms, ##n+1##, results in ##n+1-r## which...
  2. tellmesomething

    Divisibility and Remainder

    The book solution is to first take one 5 out 5(5^98)= 5(25^49)=5(26-1)^49 And then when we expand it using Binomial theorem we get a number which isnt a multiple of 13, we get -5 as the remainder. But since remainders have to be positive we add 13 to it (this i generalised by dividing numbers...
  3. RChristenk

    Find the ##r^{th}## term from beginning and end of ##(a+2x)^n##

    ##r^{th}## term counting from the beginning: The coefficient of the ##r^{th}## term is ##r-1## ##^nC_{r-1}a^{n-(r-1)}(2x)^{r-1} = ^nC_{r-1}a^{n-r+1}(2x)^{r-1}## This is the correct answer. ##r^{th}## term counting from the end: There are a total of ##n+1## terms in ##(a+2x)^n##...
  4. RChristenk

    Why is ##^{2m}C_m## equivalent to ##\dfrac{2m!}{m!m!}##?

    By definition, ##^nC_r=\dfrac{n(n-1)(n-2)...(n-r+1)}{r!}##. This can be simplified to ##^nC_r=\dfrac{n!}{r!(n-r)!}##, which leads to ##^{2m}C_m=\dfrac{2m!}{m!m!}##. But I can't see how from the original equation ##^{2m}C_m=\dfrac{(2m)(2m-1)(2m-2)...(m+1)}{m!}## is equivalent to...
  5. S

    Binomial Theorem - determine the term with...

    I'm sort of stumped here , do i do this? (1+3x) \left( \frac{1+3x}{1+2x} \right)^2 = (1+3x) \left( \frac{3}{2} - \frac{1}{2(2x+1)} \right)^2 (1+3x) \left( \frac{3}{2} \right)^2 \left( 1 + \frac{-1}{3(2x+1)} \right)^2 and then apply the binomial theorem formula on the squared term above...
  6. chwala

    Solve the given problem that involves binomial theorem

    part (a) ##(4+3x)^{1.5} = 2^3+ 9x+ \left[\dfrac {1}{2} ⋅ \dfrac {3}{2} ⋅\dfrac {1}{2}⋅\dfrac {1}{2}⋅9x^2\right]+ ...## ##(4+3x)^{1.5}=8+9x+\dfrac {27}{16} x^2+...##part (b) ##x≠-\dfrac {4}{3}##part (c) ##(8+9x+\dfrac {27}{16} x^2+...)(1+ax)^2 = \dfrac{107}{16} x^2## ... ##8a^2+18a+\dfrac...
  7. chwala

    Use binomial theorem to find the complex number

    This is also pretty easy, ##z^5=(a+bi)^5## ##(a+bi)^5= a^5+\dfrac {5a^4bi}{1!}+\dfrac {20a^3(bi)^2}{2!}+\dfrac {60a^2(bi)^3}{3!}+\dfrac {120a(bi)^4}{4!}+\dfrac {120(bi)^5}{5!}## ##(a+bi)^5=a^5+5a^4bi-10a^3b^2-10a^2b^3i+5ab^4+b^5i## ##\bigl(\Re (z))=a^5-10a^3b^2+5ab^4## ##\bigl(\Im (z))=...
  8. chwala

    Solve the equation involving binomial theorem

    $$(7-6x)^3+(7+6x)^3=1736$$ $$⇒(7^3(1-\frac {6}{7}x)^3+(7^3(1+\frac {6}{7}x)^3=1736$$ $$343[1-\frac {18}{7}x+\frac {216}{98}x^2-\frac{1296}{2058}x^3]+343[1+\frac {18}{7}x+\frac {216}{98}x^2+\frac{1296}{2058}x^3]=1736$$ $$343[2+\frac {432}{98}x^2]=1736$$ $$686+\frac {148,176}{98}x^2=1736$$ $$\frac...
  9. S

    B Binomial Theorem: Exploring Meaning of Coefficients in General Expansion

    In the general expansion of (1+x)^n what does the sum of the coefficients mean?
  10. D

    I Binomial theorem with more than 2 terms

    Hi. Is the binomial theorem ##(1+x)^n = 1+nx+(n(n-1)/2)x^2 + ….## valid for x replaced by an infinite series such as ##x+x^2+x^3+...## with every x in the formula replaced by the infinite series ? If so , does the modulus of the infinite series have to be less than one for the series to...
  11. E

    MHB Determine an expression using binomial theorem

    Determine an expression for f(x) =(1+x)(1+2x)(1+3x)…(1 +nx),find f⸍(0) .
  12. J

    A Newton's Generalized Binomial Theorem

    I'm trying to expand the following using Newton's Generalized Binomial Theorem. $$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$ where $$0<\delta<<1$$ But the condition for this formula is that...
  13. Schaus

    Binomial Theorem - Determine n

    Homework Statement The sixth term of the expansion of (x-1/5)n is -1287/(3125)x8. Determine n. Homework Equations tk+1=nCkan-kbk The Attempt at a Solution tk+1=nCkan-kbk t5+1=nC5(x)n-5(-1/5)5 This is where I'm stuck. Do I sub in -1287/(3125)x8 to = t6? If so what do I do from here...
  14. Schaus

    Solve Binomial Theorem Homework: Find Coefficients of Degree 17 & x7

    Homework Statement 1. Given the binomial (x2-x)13determine the coefficient of the term of degree 17. Answer = -715 2. Given the binomial (2x+3)10 determine the coefficient of the term containing x7. Answer = 414720 2. Homework Equations tk+1=nCkan-kbk The Attempt at a Solution #1 - What...
  15. Alettix

    B Binomial Expansion with Negative/Rational Powers

    Hello! When studying binominal expansion: ## (a+b)^n = \sum_{k=0}^{n}{{n \choose k}a^{n-k}b^k} ## in high school, we proved this formula with combinatorics considering that "you can choose either a or b each time you multiply with a binom". Probably, this is not a real mathematical proof at...
  16. N

    A very very hard college algebra problem

    Homework Statement Note: I'm saying it's very very hard because I still couldn't solve it and I've posted it in stackexchange and no answer till now. I'm posting here the problem statement, all variables and known data in addition to my solving attempts. Because I'm posting an image of my...
  17. N

    Expand x^(k+1)/(k+1) - (x-1)^(k+1)/(k+1)

    Homework Statement Expand x(k+1)/(k+1) - (x-1)(k+1)/(k+1) Homework Equations (a+b)m = am + mam - 1b + (mℂ2)am - 2b2 + ... + bm[/B] The Attempt at a Solution Here is my solution, I would like to know if it's correct or not I have the solution in an attached image
  18. T

    I Damped Oscillators and Binomial theorem step

    I uploaded a picture of what I am stuck on. I understand the equation of motion 3.4.5a for a damped oscillator but I don't understand how to use binomial theorem to get the expanded equation 3.4.5b. I am no where near clever enough to figure this one out. I know how to use binomial theorem to...
  19. W

    Legendre polynomial of zero

    Homework Statement Using the Generating function for Legendre polynomials, show that: ##P_n(0)=\begin{cases}0 & n \ is \ odd\\\frac{(-1)^n (2n)!}{2^{2n} (n!)^2} & n \ is \ even\end{cases}## Homework Equations Generating function: ##(1-2xt+t^2)^{-1/2}=\displaystyle\sum\limits_{n=0}^\infty...
  20. M

    MHB Solve Sum of {30 \choose i} with Binomial Theorem

    Simplify (find the sum) of {30 \choose 0} + \frac{1}{2}{30 \choose 1}+ \frac{1}{3}{30 \choose 2} + ... + \frac{1}{31}{30 \choose 30}. Do this is two ways: 1. Write \frac{1}{i+1}{30 \choose i} in a different way then add 2. Integrate the binomial thorem (don't forget the constant of integration)...
  21. sinkersub

    Inverse Binomial Expansion within Laurent Series?

    Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...
  22. Orange-Juice

    Applying binomial theorem to prove combinatorics identity

    Homework Statement Prove that \sum\limits_{k=0}^l{n \choose k}{m \choose l-k} = {n+m \choose k}Homework Equations Binomial theorem The Attempt at a Solution [/B] We know that (1+x)^n(1+x)^m = (1+x)^{n+m} which, by the binomial theorem, is equivalent to: {\sum\limits_{k=0}^n{n \choose...
  23. G

    Evaluating Finite Sum: Homework Statement

    Homework Statement Find \sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k} Homework Equations -Binomial theorem The Attempt at a Solution I am using the binomial coefficient identity {n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}: \sum\limits_{k=0}^{n}k^2{n\choose...
  24. Matejxx1

    Please ask me questions to challenge my knowledge

    So I was checking the How to self-study math thread and saw that someone suggested that It would be helpfull to create this kind of thread. And because we are writting a test on thursday on Probability I though it would be nice to find out which parts I still need to double-check. So these...
  25. G

    Use of binomial theorem in a sum of binomial coefficients?

    Homework Statement How to use binomial theorem for finding sums with binomial coefficients? Example: S={n\choose 1}-3{n\choose 3}+9{n\choose 5}-... How to represent this sum using \sum\limits notation (with binomial theorem)? Homework Equations (a+b)^n=\sum\limits_{k=0}^{n}{n\choose...
  26. W

    Interesting Probability problem and maybe binomial theorem

    Homework Statement For reference, this is the image setting up the problem. "A wireless sensor grid consists of 21×11=231 sensor nodes that are located at points (i,j) in the plane such that i∈{0,1,⋯,20} and j∈{0,1,2,⋯,10} as shown in Figure 2.1. The sensor node located at point (0,0) needs...
  27. Keen94

    Proofs using the binomial theorem

    Homework Statement Prove that ∑nj=0(-1)j(nCj)=0Homework Equations Definition of binomial theorem. The Attempt at a Solution If n∈ℕ and 0≤ j < n then 0=∑nj=0(-1)j(nCj) We know that if a,b∈ℝ and n∈ℕ then (a+b)n=∑nj=0(nCj)(an-jbj) Let a=1 and b= -1 so that 0=(1+(-1))n=∑nj=0(nCj)(1n-j(-1)j)...
  28. shanepitts

    How is the binomial theorem used here?

    The below image shows a portion of my current Analytical Mechanics textbook. My inquiry is how is the binomial theorem used to get from eq. 3.4.5a ⇒ 3.4.5b ? Thanks in advance
  29. AdityaDev

    Summation with binomial coefficients question

    Homework Statement ##\sum\limits_{r=0}^n\frac{1}{^nC_r}=a##. Then find the value of $$\sum\sum\limits_{0\le i<j\le n}(\frac{i}{^nC_i}+\frac{j}{^nC_j})$$ Homework Equations I have used two equations which I derived myself. This is the first one. The second one is: 3. The Attempt at a...
  30. A

    Binomial Theorem: 11 Terms Explained

    Any hints for this : 1- (11C1/2.3 ).2^2 + (11C2/3.4 ). 2^3 ...so on up to 12 terms .
  31. A

    MHB Solve the Binomial Theorem Puzzle: Find Missing Member

    Screenshot by Lightshot The translation in binom coefficent of 4th and 10th are mathching each other. Find the member which doesn't have x in it. I understand all of it but the part where (n up n-3)=(n up 9) I just don't understand how they got 12 here
  32. A

    MHB Solve Binominal Form (4x+3)^n | Binomial Coefficients

    In solved binominal form (4x+3)^n has two members x^4 and x^3 whose binomial coefficients are equal. I'm kinda good in solving binomial coefficient, but I never stumbled to something like this
  33. Q

    Binomial theorem and modular arithmetic

    Homework Statement From an old exam: Show that \begin{equation*} \sum_{0 \leq 2k \leq n} \binom{n}{2k}2^k = 0 (3) \text{ iff } n = 2 (4). \end{equation*} By ##a = b (k)## I mean that ##a## is congruent to ##b## modulo ##k##. Homework Equations Binomial theorem: ## (a + b)^m =...
  34. Greg Bernhardt

    What is binomial theorem

    Definition/Summary The binomial theorem gives the expansion of a binomial (x+y)^n as a summation of terms. The binomial theorem for positive integral values of 'n', is closely related to Pascal's triangle. Equations The theorem states, for any n \; \epsilon \; \mathbb{N} (x+y)^n =...
  35. anemone

    MHB Optimizing Binomial Coefficients for Maximum Value

    From the binomial theorem, we have $\displaystyle \begin{align*}\left(1+\dfrac{1}{5}\right)^{1000}&={1000 \choose 0}\left(\dfrac{1}{5}\right)^{0}+{1000 \choose 1}\left(\dfrac{1}{5}\right)^{1}+{1000 \choose 2}\left(\dfrac{1}{5}\right)^{2}+\cdots+{1000 \choose...
  36. J

    Calculating n!/(k-1)!(n-k+1)! from Binomial Theorem

    1. How do you get n!/(k-1)!(n-k+1)! from \begin{pmatrix} n\\k-1 \end{pmatrix} I thought it would be n!/(k-1)!(n-k-1)! where the n-k+1 on the bottom of the fraction would be a n-k-1 instead. I don't understand why there is a "+1" wouldn't you just replace k with k-1 in the binomial formula?
  37. A

    Exploring the Depth of the Binomial Theorem: A Scientist's Perspective

    Hello all! This isn't a problem in particular I'm having trouble with, but a much more general question about the binomial theorem. I'm using Stewart's precal book. The section devoted to the theorem has several problems dealing with proving different aspects of it, mostly having to do with...
  38. 22990atinesh

    Importance of Binomial Theorem

    I know Binomial Theorem is a quick way of expanding a Binomial Expression that has been raised to some power i.e ##(a+b)^n##. But why is it so important to expand ##(a+b)^n##. What is the practical use of this in Science and Engineering.
  39. MarkFL

    MHB George's question at Yahoo Answers regarding the binomial theorem

    Here is the question: I have posted a link there to this thread to the OP can view my work.
  40. B

    How do I expand (1 + x)^{2}(1 - 5x)^{14} as a series of powers of x?

    Hello, I have a problem regarding the binomial theorem and a number of questions about what I can and can't do. Homework Statement Write the binomial expansion of (1 + x)^{2}(1 - 5x)^{14} as a series of powers of x as far as the term in x^{2} Homework Equations The Attempt at a Solution I...
  41. MarkFL

    MHB Vandomo's question at Yahoo Answers regarding the binomial theorem

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  42. P

    MHB Simplify equation using binomial theorem

    I'm sure this is easy but it has got me baffled. I'm told that the binomial theorem can be used to simplify the following formula x = \dfrac{1 - ay/2}{\sqrt{1-ay}} to (approximately) x = 1 + a^2 y^2 / 8 if a << 1. Thanks for any help or pointers on this one in particular, and/or general...
  43. J

    Quotient rule and binomial theorem

    If it's possible to relate the product rule with the binomial theorem, so: (x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2 D^2(fg)=1f^{(2)}g^{(0)}+2f^{(1)}g^{(1)}+1f^{(0)}g^{(2)} So, is it possible to relate the quotient rule with the binomial theorem too?
  44. Saitama

    MHB Finding b_n - Binomial theorem problem

    Question: If $\displaystyle \sum_{r=0}^{2n} a_r(x-2)^r=\sum_{r=0}^{2n} b_r(x-3)^r$ and $a_k=1$ for all $k \geq n$, then show that $b_n={}^{2n+1}C_{n+1}$. Attempt: I haven't been able to make any useful attempt on this one. I could rewrite it to: $$\sum_{r=0}^{n-1} a_r(x-2)^r +...
  45. P

    Binomial theorem proof by induction

    On my problem sheet I got asked to prove: ## (1+x)^n = \displaystyle\sum _{k=0} ^n \binom{n}{k} x^k ## here is my attempt by induction... n = 0 LHS## (1+x)^0 = 1 ## RHS:## \displaystyle \sum_{k=0} ^0 \binom{0}{k} x^k = \binom{0}{0}x^0 = 1\times 1 = 1 ## LHS = RHS hence true for...
  46. paulmdrdo1

    MHB Find Term with $x^2$ in Binomial Theorem

    find the term with $x^2$ $\displaystyle\left(x^2-\frac{1}{x}\right)^{10}$ thanks!
  47. applestrudle

    Binomial theorem to evaluate limits?

    Homework Statement lim x->1 (X^9 + x -2)/(x^4 + x -2) I know how to do this using L'Hopitals Rule and I get 2 Homework Equations (1+b)^n = 1 + bn + n(n-1)b^2/2! + n(n-1)(n-2)b^3/3! ... The Attempt at a Solution Let x = h+1 x -> 1 h -> 0 lim h->0 (h+1)^9 +...
  48. MarkFL

    MHB Apply Binomial Theorem: Expand (x-2y)^3

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  49. M

    Binomial Theorem problem

    I highlighted the portion in red in the paint document that I'm not understanding. How can we see by inspection that the product is equal to the series 2?
  50. reenmachine

    Finding the Coefficient of x^6y^3 using Binomial Theorem in (3x-2y)^9

    Homework Statement Use the binomial theorem to find the coefficient of ##x^6y^3## in ##(3x-2y)^9##. Homework Equations ##1+9+36+84+126+126+84+36+9+1## (I used two lines for the lenght) ##1(3x)^9(-2y)^0+9(3x)^8(-2y)^1+36(3x)^7(-2y)^2+84(3x)^6(-2y)^3+126(3x)^5(-2y)^4##...
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