Application of Stirling's Formula

An often used application of Stirling's approximation is an asymptotic formula for the binomial coefficient. One can prove that for k = o(n exp3/4), (n "choose" k) ~ c(ne/k)^(k) for some appropriate constant c. Can you find the c? Can you say why this only works when k is much smaller than n exp3/4?

Euler Path Problem : The Seven Bridges of Konigsberg

In Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this: The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

Twenty workers are to be assigned to 20 different jobs, one to each job. How many different assignments are possible?

Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much. 2. Twenty workers are to be assigned to 20 different jobs, one to each job. How many different assignments are possible? 3. A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many if she must answer at least 3 of the first 5 questions? 4. A dance class consists of 22 students, 10 women and 12 men. If 5 men and 5 women are to be chosen and then paired off, how many results are possible?

In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each? From a group of n people, suppose that we want to choose a committee of k, k <=n, one of whom is to be designated as a chairperson. By focusing first on the choice of the committee and then on the choice of the chair, argue that there are (n choose k)•k possible choices.

5. In how many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each? 6. From a group of n people, suppose that we want to choose a committee of k, k <= n, one of whom is to be designated as a chairperson. (a) By focusing first on the choice of the committee and then on the choice of the chair, argue that there are (n choose k)•k possible choices. (b) By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are (n choose k-1)•(n - k + 1) possible choices. (c) By focusing first on the choice of the chair and then on the choice of the other committee members, ... click for more

The problems are from probability class.

The problems are from 400 level probability class but introductory course. Please specify the terms you use (if necessary) and explain each step of your solutions. Thank you very much.