Binomial Definition and 640 Threads

Homework Statement Find the coefficient of x^{29} in the expansion of (1+x^{5}+x^{7}+x^{9}).Homework Equations The Attempt at a Solution

Could anyone help me on this question? Is it true that \sum_{k=n+1}^{2n}\left(\begin{array}{c} 2n\\k\end{array}\right)x^{k}\left(1-x\right)^{2n-k}\leq2x for any x\in(0,1) and any positive integer n? Any help on that will be greatly appreciated!

Hey guys, I've been reading up on binomial coefficients and I have found a brief section on n choose r. I understand vaguely what it actually is, however in my textbook there is a step by step proof of how we show that: ( \stackrel{n}{r} ) = \frac{n!}{r!(n-r)!} I can follow where...

Homework Statement Now you and your fiend play a different game. You flip your coin until it comes up heads the first time. Let X denote the number of flips needed. Your friend rolls its die until it comes up "3" or "5". The first try let Y denote the number of rolls needed. Assume X and Y are...

Homework Statement A drug company markets a medication that cures about 60% of cases with depression. A CB program is thought to be more effective. It was delivered to 15 depressed people. Determine the minimum number of cured people required to support the claim that the CB program is more...

Homework Statement prove that (\stackrel{2n}{n}) is even when n \geq1 Homework Equations as a hint they gave me this identity: \stackrel{n}{k}= (n/k)(\stackrel{n-1}{k-1}) The Attempt at a Solution by using that identity i got: (\stackrel{2n}{n}) = (2n/n)...

Homework Statement Determine the coefficient of p4q7 in the expansion of (2p-q)(p+q)10Homework Equations The Attempt at a Solution sry, i can't attempt to solve this coz i don't even know how to expand this using formula

Homework Statement \mbox{Prove or give a counterexample: If p is a prime integer, then for all integers x and y, } (x+p)^p \equiv_p x^p+y^p. Homework Equations \equiv_p \mbox{just means (mod p). Can you please check and see if this proof is well-formed?} The Attempt at a Solution...

Homework Statement Let X and Y be independent binomial random variables with parameters n and p. Find the PMF of X+Y. Find the conditional PMF of X given that X+Y=m. Homework Equations The PMF of X is P(X=k)=(n C k)pk(1-p)n-k The PMF for Y would be the same. The Attempt at a...

Let X be a Binomial B(\frac{1}{2},n), where n=2m. Let a(m,k) = \frac{4^m}{(\stackrel{2m}{m})}P(X = m + k). Show that lim_{m->\infty}(a(m,k))^2 = e^{-k^2}. So far, I've found that P(X = m+k) = (\stackrel{2m}{m+k}) \frac{1}{4^m} Then, a(m,k)=\frac{m!m!}{(m+k)!(m-k)!}. But I have no...

Here is the problem i am having trouble: Expressing the binomial coefficients in terms of factorials and simplifying algebraically show that (n over r) = (n-2+1)/r (n over r-1) i got that equals ((n-r+1)/r) ((n!)/((r-1)!(n-(r-1))!)) but i am trying to get that to equal n!/r!(n-r)! which...

Hi, just started learning probability & need some help in understanding... "The binomial random variable X associated with a binomial experiment consisting of n trials is defined as X = the number of S's among the n trials. Suppose, for example, that n = 3. Then there are 8 possible...

Dear Reader, I am writing for information, or a point towards any information about the calculation on the uncertainty on the number of trials in a binomial distribution. I had been using the SQRT(N) (taken from poisson dist. I miss them) but forgot they are binomial. For example if I toss a...

[b]1. Let the p.m.f. pf M be defined by f(m)=x/8, x=1,3,4. What is the mean of M? [b]2. n!/n-r*p^n*(1-p)^n-r [b]3. 3!*1/3^3*2/3^2=.59 This is not the correct answer!

Hi, its me again. \left(1 - x\right)^{\frac{1}{5}} show that 31.2^{\frac{1}{5}} \approx \frac{197}{99} how can i know what value should x be ?

Hi, I'd like to know how to find the uncertainty of a function that has two binomial distribution s, something like Signal = N(yes) - N(no) Set p = 0.6 for yes. My problem is that I do not know how to find the uncertainty for N(yes) and N(no), and do not know how to find the uncertainty...

I know how to derive the binomial approximation for (1+\alpha x)^{\gamma} using a Mellin transform, but for (1-\alpha x)^{\gamma} the method appears to fail because I can't take x to infinity. Here is the basics of the method. Take the Mellin transform of (1+\alpha x)^{\gamma}: M(p) =...

Homework Statement Homework Equations Formula => C(n,r) or nCr =n!/r!(n-r)! & the basic Binomial Theorem formula. *Answer mentioned in book = nx The Attempt at a Solution The LHS should be (x+y)n & the given question is its expansion only if that 'r' is not multiplied in the question. I...

Homework Statement A third degree polynomial has 3 roots that, when arranged in ascending order, form an arithmetic progression in which the sum of the 3 roots equal 9/5. The difference between the square of the greatest root and the smallest root is 24/5 Given that the coefficient of the...

When using a binomial series to expand 1/\sqrt{1-x^{2}} I come up with the correct answer except that I do not add the number one to my answer. Why do I have to add one to the series, should this not arise when calculating the sum?

Homework Statement Suppose that the conditional distribution of X given that N = n is binomial (n, 1/2) and the distribution of N is uniform over {2,4,6} a) Determine P(X=2, N = 4) b) Determine P(X=1) c) Determine P(N = 4| X =1) Homework Equations The Attempt at a Solution...

Firstly, I want to note I'm a post college student who is attempting to teach himself calculus. I'm reading Calculus Made Easy by Silvanus P. Thompson and Martin Gardner, St. Martin's Press, 1998 ed. My question comes from page 56 Case of a Negative Exponent y + dy= (x + dx)^-2...

Hello, All we know the Binomial Theorm which may be stated mathematically as: \left(x+y\right)^n=\sum_{k=0}^n{n\choose k}y^k\,x^{n-k} Now suppose that we have the following mathematical expression: \sum_{k=0}^{n}{n\choose k}\,(-1)^k if we substitute x=1 and y=-1 in the first...

Hello! Is there a closed form expression or a good estimate for the probability that a binomial distribution yield the average np or less. Basically I'm asking for a good way to evaluate P=\sum_{k=0}^{np} \begin{pmatrix} n\\ k \end{pmatrix} p^k(1-p)^{n-k} I just figured that for the...

use the binomial theorem to determine the coefficient of x^33 in the expansion of (\frac{1}{4}-2x^3)^17 ive played around with it and come up with 33^C_17 as a coefficient.am i right in saying that is all the question asks Homework Equations The Attempt at a Solution

Hi Friends Please Solve my problem Please TEll Properly dq = Surface Charge da = Surface Area Radius of ring = w Total Radius = R qo = Point Charge Distance from Center of Ring to Point Charge = Z I want to know how I ca apply binomial theorem to solve the equation If Z = 0...

Please help with this thanks :) 1. (a) Define the Poisson probability distribution with mean μ. (b) Write down the binomial distribution for x successes in n independent trials each with probability p of success. (c) On average, 0.15% of the nails manufactured at a factory are known to...

Prove this using this identity: k\binom{n}{k}=n\binom{n-1}{k-1} \binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}+...+(-1)n-1\binom{n}{n} I was able to do this via differentiation, but not using this substitution. Any hints would be great.

\sumk=0n\binom{n}{k}2=\binom{2n}{n} Could someone give me a hint as to how to start this. I'm not sure how to really interpret it. (n-k)\binom{n}{k}=n\binom{n-1}{k} Right Side: Suppose you create a committe from \binom{n}{k} , then to pick a leader who isn't in the committee but...

\sum_{m=k}^{n-k}\binom{m}{k}\binom{n-m}{k}=\binom{n+1}{2k+1} I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.

hello, i am supposed to show that Sigma of k = 0 to m, (n, k) (n - k, m - k) = 2^m (n, m) So I have after expanding: (n, k) = n!/(n-k)!k! and (n-k, m-k) = (n-k)!/(m-k)!(n-m)! so together the (n-k)! cancels out and I have n!/k!(m-k)!(n-m)! and that is n!/m!(n-m)! which is (n, m)...

Using the Binomial Theorem and the definition of the derivate of a function f(x) as f'(x)= lim as h tends to 0 ((f(x+h)-f(x))/h) Prove that if f(x)=x^n then f'(x)=nx^(n-1) I'm confused as to how to exactly incorporate the nCr "n choose r" into this interpretation of the...

\binom{r}{k}=\frac{r}{r-k}\binom{n-1}{k} I'm having problems proving this. However, here is my reasoning: when factoring out an r you get \frac{r*(r-1)!}{(r-k)!k!} \frac{r}{r-k}*\frac{(r-1)!}{(r-k-1)!k!} Is this proper reasoning?

Hi Guys, I have been given the probability that a drill strikes oil in a region = 0.2. I know that if I wanted to find the probabilty of say striking oil 3 times out of 5 wells It would be 5Choose3 = 5!/((2!)(3!)) * (1/5)3* (4/5)2 = 0.0512 My question is how would I go about...

Hi, My question is based around the idea about calculating the number of possible outcomes when a given number of variables are chosen randomly (all with equal probability of being picked) a given number of times. Most importantly, I an specifically working so that order is redundant. ie AAB =...

Homework Statement prove, using mathematical induction, that the next equation holds for all positive t. \sum_{k=0}^n \dbinom{k+t}{k} = \dbinom{t+n+1}{n} Homework Equations \dbinom{n}{k} = {{n!} \over {k!(n-k)!}The Attempt at a Solution checked that the base is correct (for t=0, and even for...

Homework Statement (a) Expand (1-x6)4 (b) Find the coefficient of xr, where r is a non-negative integer, in the expansion of (1-x)-4 for |x|<1. (c) Using (a) and (b), or otherwise, find the coefficient of x^8 in the expansion of ((1-x6)/(1-x))4 for |x|<1. (Answers: (a)...

Homework Statement You've got N marbles, and N bins. one by one, a marble is randomly placed in a bin. What is the probability that there will be no marbles in a given bin. Homework Equations P=N!/(n1!n2!)*p^n1*q^n2 : binomial probability The Attempt at a Solution since we're...

I'm trying to figure out this problem but i keep getting stuck. Homework Statement A woman wants to have a 95% chance for a least a one boy and at least one girl. What is the minimum number of children that she should plan to have? Assume that the event that a child is a girl and a boy is...

Homework Statement The mailing list of an agency that markets scuba-diving trips to the Florida Keys contains 70% males and 30% females. The agency calls 30 people chosen at random from its list. What is the probability that the first woman is reached on the fourth call? (That is, the first 4...

Homework Statement Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently a) What is the probability...

Homework Statement Using the identity (1+2x) (1-2x+4x^2) = 1+8x^3 to expand (1-2x+4x^2)^(-0.5) in ascending powers of x as far as the term in x^3. (The answer is 1+x-x^2/2-7x^3/2+...) Homework Equations (1+2x)^(0.5) = 1+x-x^2/2+x^3/2+... (1+8x^3)^(-0.5) = 1-4x^3+... The Attempt at...

Hi, Let's say you start off with $500, and someone offers to give you another $500 everytime a coin is heads, or take $500 from you it's tails. You agree to play this game until a) you've either lost all your money, or b) you've made an extra $1000 (i.e. so you walk away with $1500). Then...

Hi. i keep finding a different answer than what textbook offers. is my answer correct? question: the quality control department of a company making computer chips knows that 2% of the chips arw defective. use the nurmal approximation to the binomial probability distribution, with a continuity...

Hello, Can someone please tell me whether I can use negative binomial distribution for this question. "If there are 3 types of books in a bookstore and each book has an equal probability of being bought. What is the expected number of purchases to get all 3 books?" Using negative...

Homework Statement A coin can be flipped a maximum of four times The following conditions exist: H(first) = $1 H(second) = $2 H(third) = $3 H(fourth) = $4 Where H = Heads And first, second, third and fourth, refer to what order one head is obtained. What is the expected gain...

Homework Statement x^2 = 2/11x + 99/121 Homework Equations The Attempt at a Solution x^2 = 2/11 x + 99/121 x^2 - 2/11x - 99/121 = 0 x^2 - 2/11x =99/121 I understand that (b/2)^2 must be added to each side to become a perfect square trinomial...But HOW I do it...

I am asked to prove that \sum ^n _{k=0} (C^n_k)^2 = C^{2 n}_n Where C^n_k signifies "n choose k" I am told the hint to use the binomial theorem and to calculate the coefficient of x^n in the product (1+x)^n (1+x)^n = (1+x)^{2n} the Binomial theorem is given by (x+y)^n = \sum_{k=0}...

Homework Statement Estimate the probability that, in a group of five people, at least two of them have the same zodiacal sign. (There are 12 zodiacal signs; assume that each sign is equally likely for any person.) Homework Equations P(X=k) = nCk * p^{k} * (1-p)^k{} The Attempt at a...

Hi, I've been struggling with this problem for sometime. Let nCk be the kth coefficient in the binomial expansion of (a+b)^n. Find an expression for (n+1)Ck in term of the various nCj. Feel free to treat k=0 and k=n+1 as special cases.