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(COMP170)[2010](s)midterm1~915^_10166.pdf
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HKUST C Department of Computer Science and Engineering
COMP170: Discrete Math Tools for CS C Spring 2010 Midterm Examination 1
Date: Thursday, March 11, 2010 Time: 19:00C20:30 Venues: LT A,B,C
Instructions
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This is a closed book exam. It consists of 17 pages and 7 questions.
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Please write your name, student ID, email, lecture section and tutorial on this page.
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For each subsequent page, please write your student ID at the top of the page in the space provided.
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Please sign the honor code statement on page 2.
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Answer all the questions within the space provided on the examination paper. You may use the back of the pages for your rough work. The last three pages are scrap paper and may also be used for rough work. Each question is on a separate page. This is for clarityand is not meant to imply that each question requiresa full page answer. Many canbe answered using only a few lines.
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Unless otherwise speci.ed you must always explain how you de-rived your answer. A number without an explanation will be considered an incorrect answer.
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Solutions canbe written in terms of binomial coe.cients and falling facto-
rials. For example, 35 + 24 may be written instead of 16, and 53 instead of 60. Calculators may be used for the exam (but are not necessary).
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Please do not use the nPk and nCk notation. Use nk and nk instead.
Questions 1 2 3 4 5 6 7 Total
Points 15 20 10 13 14 16 12 100
Score
As part of HKUSTs introduction of an honor code, the HKUST Senatehas recommended that all students be asked to sign a brief declaration printed on examination answer books that their answers are their own work, and that they are aware of the regulations relating to academic integrity. Following this, please read and sign the declaration below.
I declare that the answers submitted for this examination are my own work.
I understand that sanctions will be imposed, if I am found to have violated the University regulations governing academic integrity.
Students Name:
Students Signature:
Problem 1: [15 pts] A by-election is being held to .ll 5 seats in a council that have recentlybecome vacant. A totalof9 candidates have registeredto run in the election. The candidates can be divided into 3 groups based on their political a.liations with: 4 in Group A, 3 in Group B, and 2 in Group C.
(a)
In how many ways can a voter choose 5 candidates?
(b)
In how many ways can a voter choose 5 candidates such that at least 3 candidates from Group A are chosen?
(c)
In how many ways can a voter choose 5 candidates such that at least 1 candidate from each group is chosen?
Ans:
(a)
59 = 126.
(b)
Wepartition this into twocases: choosing either3 or4,fromgroupA and, respectively, 2 or 1 candidates from the combiniation of groups B and C. This is
... . ... .
45 45
+ =4 10+1 5 =45.
32 41
(c)In thiscaseweneed topartition intoallthepo