The University of Melbourne — School of Mathematics and Statistics
MAST30012 Discrete Mathematics — Semester 2, 2021
Practice Class 1: Direct Enumeration – Answers
Q1: Hint: Partition the problem on squared distance from origin. Answer: 21.
Q2: Hint: Partition according to the value of a 2 {1, 2, . . . , 100}.
Use that
nP
k=1
k
2 =
1
6
n(n+ 1)(2n+ 1). Answer:
1
6
99 · 100 · 199.
Q3: There are 18 possible rolls of two dice with an even sum. Hence the probability is 1/2.
Q4: There are 368 � 268 passwords containing at least one digit.
Q5: Answers: (a) 75. (b) (7)5 = 7 · 6 · 5 · 4 · 3. (c) 7 · 64. (d) 462 =
�
5+7�1
5
�
.
Q6: Let n be the number of digits in the bit sting. There are 2n/2 palindromes when n is even
and 2(n+1)/2 when n is odd (can be expressed as 2dn/2e for any n).
Q7: Hint: Place a blue car between each red card then park remaining blue cars. Answer: 15.