CS计算机代考程序代写 discrete mathematics The University of Melbourne — School of Mathematics and Statistics

The University of Melbourne — School of Mathematics and Statistics

MAST30012 Discrete Mathematics — Semester 2, 2021

Practice Class 3: Multinomials and Inclusion-Exclusion – Answers

Q1: (a)

7

3


= 7!

3! 4!
= 7 · 5 = 35.

(b) See lectures.

(c)

10

3,3,1,2


= 10!

4! 3! 1! 2!
= 12600.

(d) The multinomial coe�cient

n
k1,k2,n�k1�k2


counts the number of ways to choose n

elements such that k1 goes into box 1 (labelled s) k2 goes into box 2 (labelled t) and the
remaining n� k1 � k2 elements goes into box 3 (labelled 1).

Q2: (a) Make a bijection such that an element xj from the set is associated with box j and
place a block into box j each time the element xj is selected.

(b)

8+4�1

4


= 11!

4! 7!
= 11⇥10⇥9⇥8

4!
= 11⇥ 10⇥ 3 = 330.

Q3: (a) Total = 11

(b) Latin only = 1

(c) Russian only = 3.

Q4: (a)


6

2, 2, 2


=

6!

2!2!2!
= 90.

(b)

(i) # (arrangements with one pair always in succession) =


5

2, 2, 1


=

5!

2!2!1!
= 30.

(ii) # (arrangements with two pairs always in succession) =


4

2, 1, 1


=

4!

2!1!1!
= 12.

iii) # (arrangements with three pairs always in succession) =


3

1, 1, 1


=

3!

1!1!1!
= 6.

(c) #(arrangements with a no pairs of socks in succession) = 30.

Q5: (a) |Pk| = 2! = 2.
(b) Pj \ Pk is the set of permutations where both j and k are fixed, so |Pj \ Pk| = 1 .
(c) |P1 \ P2 \ P3| = 1.
(d) Just plug in from above.

(e) Pr


no one collects

correct umbrella


=

1

3
.