Q’t a Treat pair each as a single object giving 5
objects Since left a night are distinct
2 5 ways ie permute r re baby g gz
T2
b a Choose ref first 23 ways
choose team Ff
23 2 ways 16224936
choose team 2,3 then Mf Y
12 3,3 ways f sanae
e Permute DI BT with gaps di la les ly
DIAOBAOT 4 perms
then choose 4 of 5 gaps
4 y ways
de
d Outcome set DX 23 a 56
I O
Z O b
3 o o co
4 o o o
5 o o o o
6 o y o G o
i 21 events Tej Probltij Iz
i prob z21
QI a K Yc choose size k team from n set then choose
captain from team
n YI choose captain from n at thee choose
K teaux from remaining en n sleet
b i RE
o
du the set of a ship binomial putts
with k east steps
Dr 623
1 if pi EMap Tire Nr Tfp pn bib bn bi fo offense
ii r Ii lie the BYC
Bonk setof instep bin paths
with k eaststeps one marked
Ip set of tuples 1 E Es with an entrymarket
I if pi ET b re Pcp pn bi ibn bi E if pi E
0 others
checkid n a 114 442 t3 5 4144 r
G t IZ t 12 tu 32 4.23 4.8 32
ko i L 3 n I z
c Chel I t 3 51 7 16 k
n 4
Prof HIIIF
I 1
a Gat
B
B
IBD IRI 14 1191131 6
IGI Il IRAH 5 IRAGABI 3
IBI 41 113A 1 5
IRUBOC t 14tlltl.to G 5 5 3
36 16 3
23
b This Q might have been interpretedin two ways Bothi ai
hold simultaneously 012 i aid are separate questions
either interpretation is correct
R G B Rink BinB
www
R G B
l
C in C
All ways 3 RinRo Gin C I
Rin R 2 ways Rin R BinB 1
Bin B 2 way fine BinB I
G n G i 2 way R C Bin same 1
ht R setof config with Rin box R
B l B in boxB
C G inbox G
1121 2 1131 2 IE1 2 j i RA131 1RAGI IBNEI1 RABAH 1
ways to have at least one bull in some colour
IRI 1 14 1131 21 2 2 I I I t 1 4
3 4 2 ways to have none
in some colour
duds B RG GBR
Q4 a S In k peonsof Cns withexath k cycles
way to seat n people around k tables
up to cyclic permutation with at least I peism
per table
b To get S luck either
Add person n between each of the n i
seated people for each possible seating
n i waysto add to 5 n i k seatings
Add a new table with seats person n alone
1 way to add to 5 n r k sealings
Boundary SCn o 0 no ways toseat is at zerotables
SCnn 1 one way to seat n at
n tables
C h 3 123 I 32 312 213 231 321
1 4G c 23 132 i 2 3 123 i3 2
S zo l SC3 1 2 S 3 2 3 SC3,3 1
QI a R lab Smallest 2 oolourd kn such that there is certainly
atleastone with a redka subgraph
and at least one blue Kb subgraph in every colouring
b Th Rin y n
Prof Casey Colour km blue no blueku or red Kz Ryu n i
Cau n Colour kn blue blue ku
colour Kn with at leastone red edge Kz
exists
Ra n n
a animals
all points inside 0 are within dislane 2
of
o
partition hexagon
o o
o
PHP at least one
s contains
at least 2 pts
Inside polygon not
on outside edges
two pts cannot
be at two corners
of some D
with distance 2
O l O 1 Id
0 I
Spanes lemma different labelled subintervals isodd
Con t Cbd 2kt
Add n pts nti subintervals
S 1,1 so Oso
s So t KH htt
to show oil 11,01 1
Induction o i i ar o o i tried
Assume true for n i pts minting
oil k 1,0 k i
Q1 11,0
Add pt a 0
Case 0 I O 1 1 801
f
before
Cas 0 I I 8 0
Case l l O 8 0
Case I I I 8 0
Case O O O 8 0
before
O 0 1 S
1 o O S
1 F I 8 0
True for all cases of n pts
Q6 i 129 2 7anti au 9 95 2I
2m90Nantz fxnau i Ioanneo
12 x an 7x an air C4 of
12C xoao ka t G 7 x xoao t7x C God
12 2 2x G 14k t7xE Ex
C 12 7 at x2 24 10K
C 241
12 7kt x2
ii 12 7 12 3 a 4 x
24 10K
12 cc I
24 102 A 424 133 x
01 4 64 B 2 3 54 A
E EE 5 stood 4 44
51 51 61 GOLF3 a
an EgLgn 16 n n 10
b x2 3 2 Eisenbn ite
mo
E antibn 3 xnHbn 2 xnb Itd
no
a bn z 3,2 Knbr t2Exnbn It x
1 07 2b I 3 bo 42
x 3bo 2b I b i’z It 3 E E
xn bn z 3bn t2bn o
07 a
b ht Pn set of all pairingsof n board
1pm Dn
Rc Putz Ir Put UDn
T err ok
T b bDo if bEPn
biz if been
17 Well define T b E Pntc i case adds cell but
ritz
pavin
use add diwanto n
ntc board
c x Putz xtpn EachPn
Ich Pn a or Pn tape
1 z
xD e2Pz tp xfxP t p t x2p
x 2 2 P sc x t P touts
P xi I t’t t
I X XL
l se od tht
Z
3 I t PG t 1
I se x Z
d it P ng Io 7 aznar
see lecture notes
0,8 a 23145 7 24531
I
a i o 12345 o 12435
23145
ii or 31245
iii 2314502453 I 134,74
iv is 24531
3 IF 1,5 2,5 12,4 3,51641
4 4
5 t 4,5
6
v squat fi 1
I s o sis
b sign x parityof trans’t If a t o oh a f no orm
of t o oh o r o orm parityofGop
partial party
0117 EIHitti
a I 23 4 56 7 1623457
I 23 4 56 7 56
I 2 341 7 45
I 27,6457 34
I 2 6 3457
1623457 23
1623457 1236567 o 56 o 45 o 34 o 23
panty m2 even
b i
g
X
X 4
ex E
5 4 3 s r
i EEE
c
Sgsu 535251
ST zS
Su Sz Sz535
szzs so z.gs
nm AA Ar A A
odds Et AE h At
achoo 9 a 9
b bijection
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