What is Binomial theorem: Definition and 138 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),






















{\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




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




{\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




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




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

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    B How to interpret Pascal's Triangle for negative numbers?

    This answer shows an extended version of Pascal's Triangle that works for negative numbers too. In This video, Sal shows how to interpret the members of Pascal's Triangle as the sum of all the possible paths to get to that member. Is there any way we can use this same 'sum of all the possible...
  2. 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...
  3. 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...
  4. 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##...
  5. 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...
  6. 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...
  7. 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...
  8. 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))=...
  9. 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...
  10. 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?
  11. 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...
  12. 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) .
  13. 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...
  14. Schaus

    Binomial Theorem - Determine n

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  15. 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...
  16. 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...
  17. N

    A very very hard college algebra problem

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  18. 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
  19. 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...
  20. W

    Legendre polynomial of zero

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  21. M

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

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  22. sinkersub

    Inverse Binomial Expansion within Laurent Series?

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  23. Orange-Juice

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  24. G

    Evaluating Finite Sum: Homework Statement

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  25. Matejxx1

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  26. G

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  27. W

    Interesting Probability problem and maybe binomial theorem

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  28. Keen94

    Proofs using the binomial theorem

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  29. 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
  30. AdityaDev

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  31. A

    Binomial Theorem: 11 Terms Explained

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  32. 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
  33. A

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

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  34. 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 =...
  35. Greg Bernhardt

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  36. 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...
  37. 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?
  38. 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...
  39. 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.
  40. 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.
  41. 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...
  42. 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.
  43. 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...
  44. J

    Quotient rule and binomial theorem

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  45. Saitama

    MHB Finding b_n - Binomial theorem problem

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  46. P

    Binomial theorem proof by induction

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  47. paulmdrdo1

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

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  48. applestrudle

    Binomial theorem to evaluate limits?

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  49. 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.
  50. M

    What is the solution to the Binomial Theorem problem highlighted in red?

    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?