k n n

n k

n ⋅ n ⋅ ... ⋅ n

k

n ,n , ...,n n + n + ... + n

1 2

k

1 2 k

1 2 k 1 2 k

I, 2,3 , 4,6 ,

8, 12,16

,.

. .

too loaglloo lazy

48--24.3 2's's*factors ( 481--5.2=10 oeaE4 osbel

9=0,112,3 ,4 6=0

,I

A,B

∣A ∪B∣ = ∣A∣ + ∣B∣ − ∣A ∩B∣

x

50 = 40 + 25 − x

⇒ x = 15

-

-

0

3 : 3. I,

3.2,

- - - 3.333 1333J

g- : 5- I,

5- 2,

-- - . 5

- 200 1200J

15 : 15-1,15-2 ,. - - - 15 .

t.gg,

166J

Total = 333+200-66=14677

k n

P (n, k) =(n− k)!

n!

n ! = n . ( n - i) .fr - 2) .

. -

. 2 ' l

t

5-I-32--1 = 5 !

Eee =f÷=(5-3) !

3! = 6

perms of"

DOG"

3-et= 3 !

" an T\

AaRp! tivide byr¥ " Is tony: chars.

M I

1111x 11 ! ✓

A chars ) !

ssss 4-

4 ! 41.2 !" " °

÷q¥ tape

;repeats off

BB

AyBB⑦AE5 ! a 060

Y I - ' '

0617

I 2 ! ' ' ODG" G O

c i G DO

treat DOG Followup : Howmany times do

as a single D,

0,

G appear together,

character not necessarily

j !

←" that order ?

-3 !

2 !

' .

I= 4 ! IIs -

- 4 !

3! 3!

2!5!

2!5!

2!3!5!

±±e=s÷

5¥,

-

-to

I

k n

=

n k

= k n− k

(k

n)(n− k)!k!

n!

(kn) (

n−kn )

in:D

(f) k ! =P ( n,

K )

" modern NBA"

(7)LB

,KB

,MT

guard : LB

10 9 8 forward : KB

-- - n

center : MJ

guard forward center

different!! guard : MJ

forward : LB= ( 'f) 3 !

center : KB

(38)(59)

(38)(

59)

→Total

: ( Y)

1) 4 or 5 or 6 a 7 or 8

2) complementary counting" Total - (O or I or 2 or 3)

k 8 − k

ways(at least 4 b) = ways(4 b) + ways(5 b) + ways(6 b) + ways(7 b) + ways(8 b)

= + + + +(48)(

49) (

58)(

39) (

68)(

29) (

78)(

19) (

88)(

09)

Direct

f- - - - -

T T T T

a¥E⇒.

.

- c :X :L - I:X'D

HI i:*) ( If) 43=48-4,4--40

(4) (4) ( I3) 4

'c- suits of had

48.44403!

faceoff . stiff.

tHa:&

,d

O

a .!ng"'face .ae , .

' l ' Hi 4--13.48

A - 82- 6

3- 7

4- 8

9. 4.4 . 4. 4.4=9.45 s - 9

6 - 10

7- - J8 - On

q - K

9. 4=36Effigies

k¥4 'D

fatuous

∗ ∗ ∣ ∗ ∗ ∗ ∗ ∗ ∗∣ ∗ ∗ ∗ ∗

∗∣∣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

2 6 4

f- , fa f ,

t , I ' f,

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∣∣

= =

n k

k k − 1

12!2!14! ( 2

14) (1214)

(k − 1

n+ k − 1)

(bars

stars + bars)

12 stars

2 bars

-

bars = bins - I

( stash:D --

x + x + x + x + x = 251 2 3 4 5

(429)

Xi 20

distributing 25 candies to 5 friends

("EY -

- (" If )

x ,x ,x ,x ,x ≥ 0 x > 0 x ≥ 1

x = x − 1 x ≥ 0

x + x + x + x + x = 25

x + x + x + x + x = 25 − 5 = 20

x = n

1 2 3 4 5 i i

i′

i i′

1 2 3 4 5

1′

2′

3′

4′

5′

( 424)

∑i=1k

i (k−1n−1)

- --

Xi Z I

Xi - I Z O

-- 5 - 5 lx.IN

d

Ino

ie( 2044 )

PILE

x + x + x = 12

= = 55

12 − 3 = 9

1 2 3

(3 − 112 − 1) (

211)

( 29+2)

-

( II! ) from last slide.

@

stars = 9

bars =L

Eri-

- nigh Ihtt) - !%g=4"I¥TI :L a

( T) 2 :3. (2)

stick a.

Hockey I .

.

dH±s. . C 44

§ a a o

-¥ "

.

::

. ( I )£e

o o ot

1 - natural # s

(F) tri near-

symmetric

sum of It 2T . . .th = (

htt) ← * s