MATH1061/7861, Mon 19 Oct 2020
Assignment 4 is out, due Friday 30 October!
Question 1. In Eurovision:
◦ there are 18 countries in the first semifinal, and 10 of these will progress to the final;
◦ there are 18 countries in the second semifinal, and 10 of these will also progress to the final;
◦ 6 additional countries have a guaranteed place in the final without having to compete in the semifinals at all.
How many possible running orders are there for the final? iouimE9QmEEa T.mu
1
2
f
L
whee is in the free what ode
Isis
semi Z
E 726
FinalsGeo E second semi
Ceo
semi
fDecidepositions oooo Decide remainig letter
ou
D GD
Qu
Ansi 255 4.253 3.25
26
Question 2. How many passwords can you create from 5 upper-case letters, if you insist that every time you use the letter Q
you immediately follow it with the letter U? Allow
QUACK 1 au EQUUS 32 ceo
O
Oe
No Q’s at all
25.25 25.25 25
f Decide position o Decideother3
Quiceu
MANGO 30 QU
au letters
ceo
cee
or
au
first semi
I
extras
9828200
Ct 25 25.25
wow
Y
Z Bga
Question 2a. You make a password from five upper-case letters, each chosen at random.
What is the probability that, every time you use the letter Q, you immediately follow it with the letter U?
pre Eid 982277 9 8,7 7
0.827
Generaty functions solving recurrences
Question 3. If a committee of 5 is chosen at random from a group of 25 mathematicians and 25 physicists, what is the probability that at least one mathematician will be chosen?
t.IE
Question 4. What is the probability that a randomly chosen 7-element subset of {1,2,…,21} contains at most one even number?
Eae
my
III no f oheK_ l ignore every object
Pr
no Inn
Question 5. What is the probability that a random permutation of {1,2,…,9} has 5 not next to 6?
I
ni
MN n
If
7 7
n
n1
seetodag.su
Question 6.
169 End i
aim
it
Gets 7
i.EC scig7
oo
awaY9Ym9 3
94 94
◦ What is the coefficient of x3 in the expansion of (x + 9)7? ◦ What is the coefficient of x8 in the expansion of (3 − x)12? ◦ What is the coefficient of x2 in the expansion of (3x − 2)5?
Thei 3 Ion y Coefficient
i
iii
I sc
8i i
151 is f Tomwithx2
s
4434Ca
Gettin n.s.s.si si
ic
E 3 51 43
i see nsan Mr
poet
iz
x3 is 35.94
7j.kz
Mr
99 720
Question 7. Combinatorial proofs! Ben will write the questions here during the lecture.