There are 4 mathematicians m1;m2;m3;m4 and 4 computer scientists c1; c2; c3; c4. mi and ci...

There are 4 mathematicians m1;m2;m3;m4 and 4 computer scientists c1; c2; c3; c4. mi and ci are enemies for each i = 1; 2; 3; 4 (i.e. m1 and c1 are enemies, m2 and c2 are enemies etc.). By the end of part (d), we ought to know how many ways there are to line up these 8 people so that no enemies are next to each other.

(a) How many ways are there to line up the 8 people with no restriction?

(b) How many ways are there to line up the 8 people such that m1 and c1 ARE next to each other? Hint: there are 2 ways to arrange m1 and c1 between themselves. Then once we have done that, we can imagine them as “glued together". So there are now 7 objects to permute (6 people and 1 glued pair).

(c) How many ways are there to line up the 8 people so that m1 and c1 ARE next to each other AND m2 and c2 are next to each other? Hint: use the gluing idea again.

(d) For i = 1; 2; 3; 4 let Ai represent the set of permutations of the people where mi and ci are next to each other (note in part (b), you found |A1|. Use inclusion-exclusion to find the number of bad permutations |A1 U A2 U A3 U A4|. Then conclude the number of good permutations.

Solutions

Expert Solution

Colby Messinger answered 9 months ago

Next > < Previous

Related Solutions

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5 As a mathematical function, does U have a maximum...

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5 As a mathematical function, does U have a maximum or minimum value? What values of Ci correspond to the minimum value of U? What values of Ci correspond to the maximum value of U? Do these values of Ci make sense from an economic standpoint? Now let us connect the idea of economic utility to actual dollar values. To keep the values more manageable, we will use household income rather than the entire...

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5 As a mathematical function, does U have a maximum...

Calculate the values "c1, c2, c3, c4, c5" with superposition 6(c1) – (c3)=50 -3(c1) + 3(c2)=0...

Calculate the values "c1, c2, c3, c4, c5" with superposition 6(c1) – (c3)=50 -3(c1) + 3(c2)=0 9(c3) – (c2)= 160 -(c2) – 8(c3) – 2(c5) + 11(c4)=60 -3(c1) – (c2) + 4(c5)=10

Consider the following cash flows: C0 / C1/ C2 /C3 /C4 ? $ 27 / +...

Consider the following cash flows: C0 / C1/ C2 /C3 /C4 ? $ 27 / + $ 24 / + $ 24 / + $ 24 / ? $ 46 a. Which two of the following rates are the IRRs of this project? (You may select more than one answer. Single click the box with the question mark to produce a check mark for a correct answer and double click the box with the question mark to empty the box...

Cash flows of Project A and B are as following: Project C0 C1 C2 C3 C4...

Cash flows of Project A and B are as following: Project C0 C1 C2 C3 C4 C5 A -9000 2000 3000 4000 5000 6000 B -11000 2000 3000 4000 5000 6000 Compute the payback periods for the following two projects. Payback period for A = _________ years; for B __________years If the discount rate is 10%, what is the discounted payback period? Discounted payback period for A=_________ years; for B __________years

Consider the following information: Cash Flows ($) Project C0 C1 C2 C3 C4 A –6,300 2,300...

Consider the following information: Cash Flows ($) Project C0 C1 C2 C3 C4 A –6,300 2,300 2,300 2,000 0 B –1,600 0 1,000 3,300 4,300 C –3,700 2,300 1,000 1,800 1,300 a. What is the payback period on each of the above projects? (Round your answers to 2 decimal places.) Project Payback Period A year(s) B year(s) C year(s) b. Given that you wish to use the payback rule with a cutoff period of two years, which projects would you...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,000...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,000 2,000 0 0 0 0 B −4,000 2,000 2,000 5,000 2,000 2,000 C −5,000 2,000 1,300 0 2,000 2,000 a. If the opportunity cost of capital is 12%, which project(s) have a positive NPV? Positive NPV project(s) Project A Project B Project C Projects A and B Projects A and C Projects B and C Projects A, B, and C No project b. Calculate...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,500...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,500 2,500 0 0 0 0 B −5,000 2,500 2,500 5,500 2,500 2,500 C −6,250 2,500 2,500 0 2,500 2,500 a. If the opportunity cost of capital is 9%, which project(s) have a positive NPV? Positive NPV project(s) Project A Project B Project C Projects A and B Projects A and C Projects B and C Projects A, B, and C No project b. Calculate...

Consider the following information: Cash Flows ($) Project C0 C1 C2 C3 C4 A –6,200 2,200...

Consider the following information: Cash Flows ($) Project C0 C1 C2 C3 C4 A –6,200 2,200 2,200 2,900 0 B –1,500 0 1,000 3,200 4,200 C –3,800 2,200 1,300 1,700 1,200 a. What is the payback period on each of the above projects? (Round your answers to 2 decimal places.) Project Payback Period A year(s) B year(s) C year(s) b. Given that you wish to use the payback rule with a cutoff period of two years, which projects would you...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,800...

Consider the following projects: Cash Flows ($) Project C0 C1 C2 C3 C4 C5 A −2,800 2,800 0 0 0 0 B −5,600 2,800 2,800 5,800 2,800 2,800 C −7,000 2,800 2,500 0 2,800 2,800 If the opportunity cost of capital is 12%, which project(s) have a positive NPV? Positive NPV Projects Calculate the payback period for each project. Project A years Project B years Project C years Which project(s) would a firm using the payback rule accept...

Subjects