Binomial Definition and 640 Threads

[SIZE="4"]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. [SIZE="4"]Equations The theorem states, for any n \; \epsilon \...

Homework Statement Find the coefficient of x^n in the expansion of each of the following functions as a series of ascending powers of x. \frac{1}{(1+2x)(3-x)} Homework Equations The Attempt at a Solution (1+2x)^{-1} = 1 + (-1)2x + \frac{(-1)(-2)}{2!}(2x)^2 +...

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...

Expand the following functions as a series of ascending powers of x up to and including the term x^3. In each case give the range of values of x for which the expansion is valid. (1+(1/x))^(-1) The Attempt at a Solution 1 + (-1)(1/x) + (-1)(-2)(1/x^2)/2 + (-1)(-2)(-3)(1/x^3)/3! = 1 -...

Homework Statement The Binomial Distribution - already developed by Jacob Bernoulli (in 1713), et alii, before Abraham de Moivre (1667-1754 CE), et alii, developed the Normal Distribution as an approximation for it (id est, the Binomial Distribution) - gives the discrete probability...

I am puzzled by the following example of the application of binomial expansion from Bostock and Chandler's book Pure Mathematics: If n is a positive integer find the coefficient of xr in the expansion of (1+x)(1-x)n as a series of ascending powers of x. (1+x)(1-x)^{n} \equiv (1-x)^{n} +...

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?

please refer to the second line of solution, since we only concerned about the probability of getting number (5) , then why can't I just just say P=(5/6)^5 , why should I times =(5/6)^5 with (1/6)^2 ?

Show that $$\sum_{j=1}^{j=n}\binom{n}{j} \frac{(-1)^{j+1}}{j} = 1 +\frac{1}{2} +\frac{1}{3} + \cdots +\frac{1}{n}$$

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...

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.

Problem: Evaluate $$\mathop{\sum \sum}_{0\leq i<j\leq n} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ Attempt: I wrote the sum as: $$\sum_{j=1}^{n} \sum_{i=0}^{j-1} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ I am not sure how to proceed from here. I tried writing down a few terms but that doesn't seem...

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Homework Statement g(x) = \sum_{n=0}^\infty \binom{k}{n} x^n g'(x) = k\sum_{n=0}^\infty \binom{k-1}{n} x^n prove that (1+x)g'(x) = kg(x) The Attempt at a Solution k(1+x)\sum_{n=0}^\infty \binom{k-1}{n} x^n distribute k[\sum_{n=0}^\infty \binom{k-1}{n} x^n +...

Homework Statement Expand ##(1+x)^(1/3)## in ascending powers of x as far as the term ##x^3##, simplifying the terms as much as possible. By substituting 0.08 for x in your result, obtain an approximate value of the cube root of 5, giving your answer to four places of decimals. Homework...

Homework Statement My math textbook is currently on the Binomial Series now, after completing the Binomial Theorem (no problems with that one). I believe most of my trouble comes from the book's rather glancing explanation of it, only giving examples of the form ##(1 +/- kx)##. Now have...

Homework Statement Suppose that particles of two different species, A and B, can be chosen with probability p_A and p_B, respectively. What would be the probability p(N_A;N) that N_A out of N particles are of type A? The Attempt at a Solution I figured this would correspond to a binomial...

Evaluate $\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}$.

Homework Statement Question: Find the number of trials needed to be 90% sure of at least three or more success, given that probability of one success is 0.2 Homework Equations N/A The Attempt at a Solution My initial attempt at the problem was finding the probability of at least...

The average if the binomial distribution with probability k for succes is simply: <> = Nk So this means that if <> = 1 the distribution function must be peaked around 1. In general when is it a good approximation (i.e. when is the function peaked sufficiently narrow) to say that the...

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Homework Statement The value of ((^n C_0+^nC_3+...) - \frac{1}{2} (^nC_1+^nC_2+^nC_4+^nC_5+...))^2 + \frac{3}{4} (^nC_1-^nC_2+^nC_4-^nC_5...)^2 The Attempt at a Solution I can see that in the left parenthesis, the first bracket contains terms which are multiples of 3 and in the second...

Suppose I have 6 die and toss them. The probability to have n 6's is binomially distributed with parameter 1/6. Now suppose instead tossing the 6 die and having 1/6 probability for a 6 each dice's probability to show 6 grows continously in the time interval t=0 to t from 0 to 1/6. Can I then...

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...

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Homework Statement Here are the problems: A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the...

Homework Statement Prove by induction that for any positive integers a, b, and n, (a choose 0)(b choose n) + (a choose 1)(b choose n-1) + ... + (a choose n)(b choose 0) = (a+b choose n) Homework Equations (x choose y) = (x!)/((x-y)!y!) The Attempt at a Solution I am able to do the...

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...

Homework Statement Hester suspected that a die was biased in favour of a four occurring. She decided to carry out a hypothesis test. When she threw the die 15 times, she obtained a four on 6 occasions. Carry out the test, at the 5% level, stating your conclusion clearly. Homework...

This is not a homework problem. Just a curiosity. But my statistics is way rusty. Suppose a binomial probability distribution with probability p for a success. What is the expected number of trials one would have to make to get your first success? In practice, this means if we took a large...

Homework Statement I made this question for myself to try to see if I could use two approaches (Poisson Distribution and Binomial Distribution) to solve a problem: Someone's average is to make 1 out of every 3 basketball shots. What are the chances she makes exactly 2 shots in a trial of 3...

Homework Statement For ##n>0##, the expansion of ##(1+mx)^{-n}## in ascending powers of ##x## is ##1+8x+48x^{2}+...## (a) Find the constants ##m## and ##n## (b) Show that the coefficient of ##x^{400}## is in the form of ##a(4)^{k}##, where ##a## and ##k## are real constants. Homework...

Hello all, I just have a question which covers binomial distribution. Sally is a goal shooter. Assume each attempt at scoring a goal is independent, in the long term her scoring rate has been shown as 80% (i.e. 80% success rate). Question: What's the probability, (correct to 3...

Homework Statement Repeatdly roll a fair die until the outcome 3 has accurred on the 4th roll. Let X be the number of times needed in order to achieve this goal. Find E(X) and Var(X) Homework Equations The Attempt at a Solution I am having trouble deciphering this question...

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Homework Statement Expand (1-2i)^10 without the Binomial Expansion Theorem I know I need to put this in polar form and then it's simple from there, however, I am simply having a difficult time finding the angle. Drawing the complex number as a vector in the complex plane I get a...

Is it possible to do a binomial expansion of (x+y)^{1/2}? I tried to compute it with the factorial expression for the binomial coefficients, but the second term already has n=1/2 and k=1, which makes the calculation for the binomial coefficient (n 1) weird, I think. Any advice?

Homework Statement Show that if cosΦ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for ℓ^{-1}_o and ℓ^{-1}_i. Also, use Φ ≈ sinΦ.] Homework Equations Sorry that...

Show that $$ \begin{align*} \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{(-1)^n}{4^{3n}} &= \frac{\Gamma\left(\frac{1}{8}\right)^2\Gamma\left(\frac{3}{8}\right)^2}{2^{7/2}\pi^3} \tag{1}\\ \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{1}{4^{3n}}&= \frac{\pi}{\Gamma \left(\frac{3}{4}\right)^4}\tag{2}...

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?

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 +...

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Hi MHB, I've come across this problem and I think I've observed a pattern when I tried to solve it by using the method of comparison with some lower values of the exponents, but then I just couldn't deduce the answer to the problem because the pattern suggests that I can't. Here is the...

Hi all, I am trying to figure out if there is a pre-existing theorem and proof of whether or not each of the binomial coefficients in a binomial expansion of (a +b)^n are divisible by n, particularly in the case where n is a prime number. Has this already been asked and answered somewhere in...

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...

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

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 +...

Hello physics forum! I come to you with a binomial coefficient problem I am stuck on. Here is the question 1. Suppose an airport has three restaurants open, Subway Burger King and McDonalds. If all three restaurants are open and each customer is equally likely to go to each one, what is the...

Hey guys. So I need to know how to Binomial expand the following function \frac{1}{(1-x^{2})}. I need this because I have to work out \prod^{∞}_{i=1}\frac{1}{(1-x^{i})} for i up to 6. But I have to do it with Binomial expansion. If i can learn how to do \frac{1}{(1-x^{2})} then the rest...

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