Question

Combinatorics. (30 points total) a) A prospective student is planning to visit several universities during the application process. She decides to visit the University of Rochester, Cornell University, Standford University, MIT, and Johns Hopkins University. How many different itineraries can she arrange for flights to visit all of these universities in one long trip? (For example: University of Rochester Cornell UniversityStandford UniversityMITJohns Hopkins University) Note that rearranging the order of the cities you visit will result in a different itinerary regardless of how the cities are rearranged. b) Suppose the prospective student in part (a) can only visit three universities. How many different itineraries can she choose from now? c) A U of R student is selecting courses for the Fall 2019 semester. She decides to choose from a total of 28 courses. How many different ways can she register for classes if she intends to register for a total of 4 courses? (Hint: Is there a difference between registering for PHY141/PHL110/CSC161/ENG206A and ENG206A/PHY141/PHL110/CSC1617) d) The U of R student described in part (c) just learned that 4 of the courses were cancelled, and the professor of a fifth is retiring after this semester. She decides that she will take the course given by the retiring professer in order to not risk the course's availability in the future. How many different ways can she register for classes now?

Answer 1

SOLUTION

Back-up Theory

Number of ways of arranging n things among themselves = n! = n(n - 1)(n - 2) …… 3.2.1...............................….(1)

Values of n!can be directly obtained using Excel Function: Math & Trig FACT (Number).................................. (1a)

Number of ways of arranging n things taking only r at a time = ⁿP_r = (n!)/{(n - r)!}….........................................…(2)

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}…..................................…(3)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN(Number, Number_chosen)

[Number is n, Number_chosen is r]…………………..............................................................……………………. (3a)

Now to work out the solution,

Part (a)

Given, 5 universities and that rearranging the order would result in a different itinerary, number of possible itinerary = 5! [vide (1)]

= 120 Answer

Part (b)

Here we need to arrange 5 taking only 3 at a time. Vide (2), number of possible itinerary

= ⁵P₃ = (5!)/(2!)

= 5 x 4 x 3

= 60 Answer

Part (c)

Since rearranging the 4 courses selected among themselves does not alter the selection, this is a case of combination of 28 courses taking 4 at a time, which is given by ²⁸C₄ = (28!)/{(4!)(24)!} [vide (3)]

= 20475 [vide (3a)] Answer

Part (d)

In this situation, with 4 courses cancelled, only 24 courses are available for selection. Of these 24, the student wants to necessarily pick one particular course [the one whose Professor is retiring]. Thus, it is a case of selecting 3 out of remaining 23 course, which given by ²³C₃ = (23!)/{(3!)(20)!} [vide (3)]

= (23 x 22 x 21)/(3 x 2 x1)

= (23 x 11 x 7)

= 1771 Answer

DONE