The University of Melbourne — School of Mathematics and Statistics
MAST30012 Discrete Mathematics — Semester 2, 2021
Practise Class 11: Designs – Answers
Q1: Break the subsets into groupings of 3 lots of 4 subsets each. Each grouping is given to
a household and the detergents within the 4 subsets are compared over 4 days. The
detergents in the first subset are compared on day 1, those in the second subset on day 2
and so on.
As an extra feature the 4 subsets could be chosen so that all brands are included at least
once.
Q2: r = �
✓
v � 1
k � 1
◆
, r =
bk
v
; 3 6= 1⇥
8� 1
2� 1
= 7.
Q3: b = 10, v = 6, r = 5, k = 3,� = 2; 5 = 2⇥
6� 1
3� 1
; 5 =
10⇥ 3
6
.
Q4: Write down the incidence matrix and its complement by yourself.
The complement represents a design with parameters b = 10, v = 6, r = 5, k = 3,� = 2.
Q5: (a) Hint: Note that det(MTM) = det(MT )det(M) =
�
det(M)
�
2
.
(b) Hint: This follows from the definition of M .
(c) Hint: Use (b) and MTM = (k � �)I + �J .