Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) For Questions 12: A panel of 4 Democratic candidates, 4 Republican candidates, and 2 Independent candidates will participate in a political forum. 12.) How many working groups of 5 candidates can be formed if: (a) there must be 2 Democr

Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) For Questions 12: A panel of 4 Democratic candidates, 4 Republican candidates, and 2 Independent candidates will participate in a political forum. 12.) How many working groups of 5 candidates can be formed if: (a) there must be 2 Democr

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Consider the following problem: Three officers-a president, a treasurer, and a secretary-are to be chosen from among four people: Ann, Bob, Cyd, and Dan. Suppose that, for various reasons, Ann cannot be president and either Cyd or Dan must be secretary. How many ways can the officers be chosen?
Suppose that Jeremy is a highschool science teacher and that he is dividing 16 students in his advanced chemistry class into four groups of four for a lab activity. Each of the four lab groups has been assigned its own task to complete for the activity, but the students are not assigned specific roles within the group. How many different ways can Jeremy assign his students to the four lab groups?
Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 16.) A group of 2 professors and 5 students (3 grad students and 2 undergrads) go to a movie together. The professors want to sit together, and the grad students want to sit together. How many ways are there to seat this group in a row of 7 seats?
Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) For the rest of the problems in this section, decide whether to use permutations or combinations. 6.) From a group of 12 newly hired office workers, 4 assistants are selected. Each of 4 different managers will be assigned one of these assistants. In how many ways can the assistants be selected and assigned?
If n(A) = 5, n(B) = 6, and n(C) = 9, what is the greatest and least number of elements in each of the following? (a) AU BUC (b) An Bnc (a) The greatest number of elements is A U BUC is The least number of elements in AUBUC is (b) The greatest number of elements in An Bn C is The least number of elements is An B n Cis
Suppose there are 5 men and 8 women qualified to be members of a new committee of 6 people, includinga president, a vice-president, and a secretary.(a) How many different ways do you have to pick a president, vice-president and secretary if you need atleast 1 men and 1 women? Does the order matter? Why?(b) How many different committees can you form with at least 3 women? Does the order matter? Why?
A survey of 145 college students was done to find out what elective courses they were taking. Let A=the set of those taking art, B= the set of those taking basketweaving, and C= the set of those taking canoeing. The study revealed the following information.n(A)= 45n(B)= 55n(C)=40n(AnB)= 12n(AnC)= 15n(BnC)= 23n(AnBnC)= 2 How many students were not taking any of these electives?
Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 2.) (a) In how many ways can a hand of 6 hearts be chosen from an ordinary deck? (b) In how many ways can a hand of 6 black cards be chosen from an ordinary deck?
d. [Permutation] Solve the following problems - Find the number of words that can be formed by using the letters of the word "DEMOCRACY" where first letter will be always C and last letter will be R. e. [Combination] A group of 15 students is to be formed from 44 students of MATI00 Course. Among the students of the course there is 2 renowned batsmen, 3 spinners, 2 fast blowlers and 2 wicketkeepers. How many different ways this can be done so as always to (i) include 1 renowned batsman, 1 fast bowler, 1 spinner and 1 wicketkeeper; and (ii) exclude spinners? 2.
A computer programming team has 15 members consisting of eight robots and seven humans. A committee of exactly 7 of these team members must be put together to work on bug testing. You may leave your answer unreduced. (a)How many committees can be chosen that contain exactly four robots and three humans? (b) How many committees can be chosen that contain at most two robots? (c) How many committees can be chosen that contain at least one human?
Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 17.) A hand of 5 cards is dealt in a poker game. How many poker hands are possible with: (a) exactly 3 Jacks? (b) only one color (red or black)? (c) a full house (3 of one kind of card and 2 of another)?
Suppose we want to choose 4 objects, without replacement, from 17 distinct objects. (a) If the order of the choices is not relevant, how many ways can this be done? (b) If the order of the choices is relevant, how many ways can this be done?