Recent content by lizarton

Homework Statement Let s*(f) be the minimum number of transpositions of adjacent elements needed to transform the permutation f to the identity permutation. Prove that the maximum value of s*(f) over permutations of [n] is {n \choose 2}. Explain how to determine s*(f) by examining f...

I looked up compositions, and after some research I think you may have meant to say that we seek combinations, the number of ways of picking k unordered outcomes from n possibilities. I have not dealt with either of these terms in my class, so I apologize for not recognizing them off-hand...

Yes, the largest possible sum would be 3n. Using my formula above, the number of ways that the 3 trials can add up to n is \left(\stackrel{n+3-1}{3 - 1}\right) = \left(\stackrel{n + 2}{2}\right) Does that seem right to you? Since I'm looking for the number of nonnegative integer solutions...

The dial can stop at any arbitrary natural number 1..n. The number of regions isn't specified further than that, but there are k=3 trials that we consider. The problem asks to compute the probability that the sum of the three spins will equal n. However, your version of the problem is just a...

For the denominator, we would have n=k, right? The formula I wrote for arbitrary n has been simplified; (3-1)!=2, and I divided the entire quantity by 2^{n-1}. I'm not sure that I understand the relationship we draw here between 2^{n-1} and \frac{(n-1)!}{(k-1)!(n-k)!}. Was my estimate for n=9...

Thanks for replying! Are compositions the same as selections in this context? If so, here's my attempt, provided I understand you correctly: For n = 9 and k = 3, the total number of compositions would be 2^{8}. Then the number of favorable outcomes/compositions would be \frac{8!}{2!6!}...

I'm not even sure that I know how to form the simpler problem, but here's a shot: For two spins, there are n-1 combinations that will sum to n. (1,n-1) (2,n-2) ... (n-2,2) (n-1,1) The total number of outcomes is n^2 (I think?). Where do I go from here?

Homework Statement Consider a dial having a pointer that is equally likely to point to each of n region numbered 1,2,...,n. When we spin the dial three times, what is the probability that the sum of the selected numbers is n? Homework Equations A Theorem I believe is relevant: With...

Consider a dial having a pointer that is equally likely to point to each of n region numbered 1,2,...,n. When we spin the dial three times, what is the probability that the sum of the selected numbers is n? I have to use summations, and I'm sure binomial coefficients. I believe that this is...

Suppose that b_k = c_k - c_{k-1}, where \langle c \rangle is a sequence such that c_0=1 and \lim_{k → ∞} c_k = 0. Use the definition of series to determine \sum^{∞}_{k=1} b_k. I've done a little analysis, so I think that c_k is decreasing, since the first term is greater than the limit of the...

Also, if you wouldn't mind--(this homework is due in an hour, so this is more for my understanding)--in general, how do we know how to choose epsilon so that it will give us the result we are looking for? Is there a reason choosing epsilon'=1/2 was particularly useful for this problem?

So then, we have showed that 1/(1+a_n) < epsilon'/3 < epsilon', but is this the end of the proof? I guess my question is, when does the other epsilon come in? or is epsilon' just our other epsilon in disguise? Thank you so much! I could use a mind-reader for this professor :)

Homework Statement Use the definition of a limit to prove that lim [(1+an)-1] = 1/2 if lim an = 1. Homework Equations (\forall\epsilon>0)(\existsN\inN)(n\geqN \Rightarrow|an-L|<\epsilon) The Attempt at a Solution Let \epsilon be arbitrary. Since lim an exists, \existsN\inN such...