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
Recommended
