fee es - math.columbia.edu

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mn inm FEE ES Aigner How many words length 2 I letter 2kW A i C Z 3 3 Binomial coefficients L nchoosek number of subsets of shek of n things 6 142 ABcD_ xx x x x x xx x x xx 1 Pascal's 8 i Triangle lo t 3 I C oht I 46 4 1 13 G CEI 3 I 5 10 10 S 1 YT F tiny divide and conquer E Z E

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Page 1: FEE ES - math.columbia.edu

mn inm FEE ESAigner How many words length 2

I letter 2kWA

iC Z

3 3

Binomial coefficients

L nchoosek numberof subsets

of shek of n things

6 142 ABcD_xxx xx xxxx xxx

1 Pascal's 8i Triangle lo t

3 I C ohtI 46 4 1 13 G CEI 3I 5 10 10 S 1

YT F tiny divideand conquer

E Z E

Page 2: FEE ES - math.columbia.edu

421 6yess

1 7 12131402234 thy 2 2324340pick 2 use 1

YT E tina HH CH

RE lRzMxt moRxtESfCx

sgcm egcntd g.cn

gCnI KAgin n H Ii

Eek S2 znttz 2no

nnx 87,677,121,13

bx x3x3NNN

FEELIt 21,31 4 10

t

Page 3: FEE ES - math.columbia.edu

Er im for any power M 1

gcn It 2T the

Sgcn gchti gcn htt 7 no

and_n9nE Tiantai

81 1

Cilento

2 MID gin 2 67 tf2 ncn 2 _n n

NAHI

L overbunting2

n all permutationsof l h Coranynthings

na n D 3.2 I

21 3g 123

n 1 132

2 213

231B 2 3123.2 1 6 321

Page 4: FEE ES - math.columbia.edu

4 I nE E I a _2z6

tokeep discard

ignore order

12 13 14

Gt3 Ltg Gt3 Ltg Gt3Lte

12 43 21 43 1342 31 42 14 32 0413223 24 34

8241405644 Gt3 tog GYUto

2341 3241 24 31 4231 3421 04321

41.121.2I

1 ftp.kyZ38

K keep n k discard

87.6 5.4 3.213.2l S U 3.2 I

4 287.8317

56

ba s agummt

howmany monomials of degd inn variablesterms

n 3 X 4,2 0 0 X 4920 1

0 1 X Y Z

D 2 1544923 4 2,42

Page 5: FEE ES - math.columbia.edu

0 3 19,4323 24 24,43 22,423422 XYZ

cP'Fant'asugie ETI y f z

I 2 I 111 a It

i's I YIi.E.i.iX.lloy2ElonIy21

1 I421 6 22 9 Hoo

deg3 in Nycz 2 dividers Xl4123 balls ooo

ql.dz EY E 542 10

Xyz alot