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 wondere ...continues

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 sh ...continues

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 ...continues

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.

A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the committee must consist of at least 3 women and at least 2 men, how many different committees are possible?

11. A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the committee must consist of at least 3 women and at least 2 men, how many different committees are possible?

Combinations-Number of choices

1. Joey is having a party. He has 10 friends, but his mom told him he could only invite 6 of them. How many choices are there if a. there are no restrictions b. there are 2 brothers who will only attend if they can attend together c. there are 2 girls who each will not attend if the other one does

9. A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible?

9. A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts the blocks in a line, how many arrangements are possible?

A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room.

10. A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3 sets of identical twins are to be assigned to these 6 beds so that each set of twins sleeps in different beds in the same room, how many assignments are possible?

Let E, F, and G be three events. Find expressions for the events so that of E, F, and G:

12. Let E, F, and G be three events. Find expressions for the events so that of E, F, and G: (a) only E occurs; (b) both E and G but not F occur; (c) at least one of the events occurs; (d) at least two of the events occur; (e) all three occur; (f) none of the events occurs.