ENGG2440B Discrete Mathematics for Engineers
Fall 2019
Take Home Final Examination
Time allowed: 3 hours
Time: 4:00-7:00pm (GMT+8), December 17th, 2019
Policy:
• Full marks will be given for completely solved problem with well justification (can be provided in the
supplementary materials submitted). You will receive partial mark if you demonstrate sufficient
understanding of the question and its solution, otherwise you will receive zero mark.
• This is an open book, open-note, take-home examination. However, your work must be original
and you may not solicit or obtain assistance from or provide assistance to other people for any
specific content on the exam.
• It is not necessary to provide an exact numerical answer. For example, you may leave your answer
using the common notations of factorials, combination, permutation, etc., e.g., ‘8 × 5 × 42 ’, ‘5!’,
etc.
• Please allow sufficient time (at least 15 minutes) to scan and/or upload your solution to Blackboard.
Problem 1. (10%) Show that the transitive rule holds for small-o notation. In other words, let
f1 (x), f2 (x), f3 (x) be some positive functions. Show that: If f1 (x) = o(f2 (x)) and f2 (x) = o(f3 (x)),
then f1 (x) = o(f3 (x)).
Problem 2. (15%) We are distributing 10 identical gifts to 3 person, A, B and C. Person A must
receive between 2 to 5 gifts (inclusive); and person B must receive between 0 to 8 gifts (inclusive); and
person C must receive more than or equal to 0 gift.
How many ways are there to distribute the gifts to A, B, C? Show your calculations.
Problem 3. (20%) A computer system is protected by a password system. An allowable password
must be composed of 4 or 5 symbols chosen from {a, b, ..., z, 0, 1, ..., 9}, and it must contain at least one
symbol from {0, 1, ..., 9}, i.e., there is at least one digit.
Examples of allowable passwords are: 1aaa, acd2, x2z2a. On the other hand, passwords such as aabcd,
absda are not allowable.
(a) (10%) What is the number of allowable passwords for the system?
(b) (10%) Suppose that 138 students in ENGG2440B have chosen a distinct and allowable password
password. Meanwhile, you select your (allowable) password randomly with an equal probability,
what is the probability that your password is the same as one of your classmates? In your answer,
please specify the sample space and the event that you are interested in.
1
Final Exam
2
Problem 4. (15%) Consider distributing 5 cards (one hand) each to two person from a deck of 52
playing cards. What is the number of distributions that both person receive a ‘Four of a Kind’, i.e.,
the hand of 5 cards contains all the 4 cards of the same face value, e.g., it contains all ‘3’s from Heart
(♥), Spade (♠), Club (♣), Diamond (♦)?
Problem 5. (20%) Let n ≥ 3 be an integer. What is the number of length-n binary strings that
do not have three consecutive 1s?
Example: if n = 4, then the strings ‘1111’, 1110’, ‘0111’ are not eligible, yet the strings ‘1101’, ‘1100’,
‘0101’, etc., are eligible.
To answer the question, you may formulate a homogeneous linear recursion for computing T (n), the
number of length-n binary strings. You should also provide the values for T (3), T (4), T (5). However,
you are not required to solve the recursion.
Problem 6. (20%) Consider the following identity:
n
X
n
k
= n(n − 1)2n−2 + n2n−1
k
k=0
2
Give a combinatorial proof to the above. Hint: consider selecting a subset, S, of {1, 2, ..., n} together
with two non-necessarily distinct elements from S.
(Your combinatorial proof should be self-explanatory and the grading will be based completely on the
submitted final solution (no need for supplementary material).)
If you couldn’t find a combinatorial proof of the identity, you may try to prove it by other means.
However, doing so you will only receive a maximum of 15% for such proof.