F ก ก F

กก

: ก (Advanced

counting technique)

กก 2

!"#$

#$$%& #ก

##$$%& #ก

#&ก!'( #$$%& #ก)* ก%&

#&ก!'( #$$%& #ก)*+$, ก%&

ก, )

#&ก!'( #$$%& #ก- ก, )

". ก

กก 3

#$$%& #ก

/0!ก/0!$#&-ก!1 กก*$ก#ก +* ), .1# n +$,$ #ก

-ก * /0!ก,#-!* ,-#$$%& #ก

$ #$$%& #ก 1 {a_n} $กก1#, a_n

- $ก, !* a₁, a₂, ..., a_n-1

ก,##,1 {c_n} 2'( #$$%& #ก ก3, $ $ 1 {c_n} * ก#$$%& #กก,#

กก 4

145") f_n $ #$$%& #ก

f_n = f_n-1 + f_n-2

" f₁ = 1, f₂ = 1

1 10 $ก 145") 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

ก1#4/ก )4ก $ก1#

n! = n · (n-1)!

" 0! = 1

145"))

กก 5

##$$%& #ก

-)*#$$%& #ก-ก.1 # /0! ),

/0! ก*: $$'*6ก$6ก 100,000 -0) $%

&-!* ก* 5% , 7 .!.1#! 30 76ก+#*

/0!ก,ก: /0!1 " Leonardo di Pisa

$$ก!,!8$ก,,!8 ,-#.0%& %)%!0

#,ก,. ,-#%& +*$ ',+ 2 $$#,ก 2

ก,# ก,!8,. กก 1 ,%)%!0 .!#$$%& #ก .1#ก,$ #',+ n

$$#,ก,ก$+$,

กก 6

/0!! 9

/0!! 9 : ! 9 ก *#ก 3 กก$

',ก$,$ก #:* ก.ก-!0,*,+

3ก * ก; ก*',ก$ '**$<*',ก$',

* #ก-ก!8"',#* $3กก#,',ก$#

ก*#

-!* H_n .1#ก*!$* % 1!, ',ก$!$

.กก$*+ก=!$ .#$$%& #ก ก1# H_n

กก 7

". ก*##$$%& #ก

1..!#$$%& #ก +$* % ก!.1# n

+$,$ ,ก .!1 $ n = 6

2. $%# !8 $-!*'*-)*-,!ก *# n # ,* $ .1# -2.1#,

.#$$%& #ก ก!.1#!2++*!$ n # 3..!#$$%& #ก C_n $ C_n ก.1#ก-,#3!$

2++* กก # n+1 .1# ), C₃ = 5 "# 4 #

x₀, x₁, x₂, x₃ ก+* ((x₀· x₁) · x₂) · x₃, (x₀· (x₁· x₂)) · x₃, (x₀· x₁)(x₂· x₃), x₀· ((x₁· x₂) · x₃), x₀· (x₁· (x₂· x₃))

กก 8

#&ก!'( #$$%& #ก

-.ก!'( #$$%& #ก :8.$$ #&

-ก!). ! ก-)*ก1:1 ! ก-)*%>

$ #$$%& #ก)* ก%& ก k ก$&?#

#$$%& #ก ,-

a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k

$ c₁, c₂, ..., c_k 2.1#. c_k ≠ 0

%#,'(.ก#$$%& #ก)* ก%& .$'(# $ ก1!,# * ก#$$%& k ,

กก 9

) #$$%& #ก

#$$%& #ก ,- P_n = 1.11 P_n-1 2#$$%& #ก)*

ก%& ก 1

#$$%& #ก ,- f_n = f_n-1 + f_n-2 2#$$%& #ก)*

ก%& ก 2

#$$%& #ก ,- a_n = a_n-5 2#$$%& #ก)* ก%&

ก 5

#$$%& #ก ,- a_n = a_n-1 + a²_n-2 +$,2#$$%& #ก )* ก%& ก 5

กก 10

$กก>(% #$$%& #ก)* ก%&

'( #$$%& #ก)* ก%& $$&?2,#.

,- a_n = rⁿก,# a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k ก3, $

rⁿ = c₁r^n-1 + c₂r^n-2 + ... + c_k r^n-k

! *# r^n-k

r^k = c₁r^k-1 + c₂r^k-2 + ... + c_k-1r + c_k

ก$กก>(% (characteristic equation) ก'(+*

.ก$กก>(%#, กก>(% (characteristic roots)

กก 11

@>;'( #$$%& #ก)* ก%& ก

2

ก+$,:1

@>; -!* c₁ c₂ 2.1#. $$#, r² - c₁·r – c₂ = 0 $ก,ก ก r₁ r₂ *# 1 {a_n} 2'( #$$%& #ก a_n = c₁·a_n-1 + c₂·a_n-2 ก3, $ a_n = α

1·r₁ⁿ + α

2·r₂ⁿก n = 0, 1, 2, ... $ ^α₁

α22,#

1..!'( #$$%& #ก a_n = a_n-1 + 2a_n-2$ a₀ = 2, a₁ = 7

2..!'( 145"))

3..!'( #$$%& #ก a_n = 2a_n-1$ a₀ = 3

4..!'( #$$%& #ก a_n = 4a_n-2$ a₀ = 0, a₁ = 4

กก 12

@>;'( #$$%& #ก)* ก%& ก

2

ก:1 ก

k

@>; -!* c₁ c₂ 2.1#. c₂ ≠ 0 $$#, r² - c₁·r – c₂ = 0 $ก

%ก# r₀ *# 1 {a_n} 2'( #$$%& #ก a_n = c₁·a_n-1 + c₂·a_n-2 ก3, $ a_n = α

1·r₀ⁿ + α

2· n · r₀ⁿก n = 0, 1, 2, ... $ ^α₁

α22,#

5. .!'( #$$%& #ก a_n = 6a_n-1 - 9a_n-2

@>; -!* c₁, c₂, ..., c_k 2.1#. r^k = c₁r^k-1 - ... - c_k $ก,ก k

ก r₁, r₂, ..., r_k *# 1 {a_n} 2'( #$$%& #ก a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k ก3, $ a_n = α

1·r₁ⁿ + α

2· r₂ⁿ + ... + α

k· r_kⁿ

ก n = 0, 1, 2, ... $ ^α₁, α

2 , ....,α

k2,#

6. .!'( #$$%& #ก a_n = 6a_n-1 – 11a_n-2 + 6a_n-3

กก 13

@>;'( #$$%& #ก)* ก%& ก

k

ก:1

@>; -!* c₁, c₂, ..., c_k 2.1#. r^k = c₁r^k-1 - ... - c_k $ก,ก t

ก r₁, r₂, ..., r_t ,ก$ก:1ก m₁, m₂, ..., m_t $1 " m_i > 1 m₁ + m₂ + ... + m_t = k *#1 {a_n} 2'( #$$%&

#ก a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k ก3, $

a_n = (α

1,0+ α

1,1n + ... + α

1,m1-1n^m1-1)·r₁ⁿ + (α

2,0+ α

2,1n + ... + α

2,m2-1n^m2-

1)·r₂ⁿ + ... + (α

t,0+ α

t,1n + ... + α

t,mt-1n^mt-1)·r_tⁿก n = 0, 1, 2, ... $ ^α_i,j 2,#

7. .!'( #$$%& #ก a_n = -3a_n-1 – 3a_n-2 – a_n-3, " a₀ = 1, a₁= -2, a₂ = -1

กก 14

$กก>(% #$$%& #ก)*+$, ก%&

#$$%& #ก)*+$, ก%& $$&?2,#. ,-

a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k + F(n)

$ c₁, c₂, ..., c_k ,# F(n)+$,-)*4/ก )

ก

a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k

#,#$$%& #ก)* ก%& $

'( $กก,#.+*.ก'#ก '((% #$

$%& #ก)*+$, ก%& ก'(#+ #$$%& #ก )* ก%& $

กก 15

#&ก!'( #$$%& #ก)*+$, ก%&

1..!'( #$$%& #ก2++* a_n = 3·a_n-1 + 2n

<* a₁ = 3 *#'(+* +

2..!'( #$$%& #ก2++* a_n = 5·a_n-1- 6·a_n-1 + 7ⁿ

ก!'( #$$%& #ก)*+$, ก%& ,2

ก!'((% #$$%& #ก)*+$, ก%& ก,

ก!'(#+ #$$%& #ก)* ก%& $

*#.11'(,$#$ก

กก 16

@>;'( #$$%& #ก)*+$, ก%&

● @>; ก1!-!* {a_n-1} * ก#$$%& #ก)*+$, ก%&

a_n = c₁·a_n-1 + c₂·a_n-2 + ... + c_k·a_n-k + F(n)

$ c₁, c₂, ..., c_k ,# F(n) = (b_tn^t + b_t-1n^t-1 + ... + b₁n + b₀)sⁿ

$ b₁, b₂, ..., b_t, s ,# s +$,-),ก #$$%& #ก)*

$ .+*'((%-

(p_tn^t + p_t-1n^t-1 + ... + p₁n + p₀)sⁿ

<* s 2ก #$$%& #ก)*$:1ก m

.+*'((%-

n^m(p_tn^t + p_t-1n^t-1 + ... + p₁n + p₀)sⁿ

กก 17

". #$$%& #ก)*+$, ก%&

1..!'((% #$$%& #ก)*+$, ก%& a_n = 6·a_n-1 - 9 a_n-2 + F(n) $ F(n) = 3n, F(n) = n3ⁿ, F(n) = n²2ⁿ, F(n) = (n²+1)3ⁿ

2.ก1!-!* a_n 2'#$ .1#3$#ก n #ก

.#, a_n * #$$%& #ก)*+$, ก%&

a_n = a_n-1+ n

.!.1#ก. ),# 1×2 2×2 ก 2×n

2++*!$

a_n=

∑

k=1 n

k

กก 18

ก, )

-ก%A"ก$! #& '*%Aก3-)*!กกก*/0!"

,/0!-!*3ก .ก+*/0!3ก%% .!'(+*

!.ก.81'(, +*$#$-!*2'( /0!$* !กก ก,#ก#,!กก, ) (Divide-and-Conquer)

$$#, #&#ก,/0! n ก2 a /0!, $ n/b

$$#,#-)*-ก#$ /0!, g(n)

-!* f(n) .1#ก1ก-กก*/0! n +*#,

f(n) = a f(n/b) + g(n)

ก$ก#ก+*#, $ก#ก, ) (divide-and- conquer recurrence relation)

กก 19

# ,$ก#ก, )

1.ก*#B ก*!ก* ก.ก* $.$ *!

"-)*กก ก <*,* ก*!,ก ก !ก*

)% <*,* ก*!* ก#, ก -!*ก:1ก)* $ 8ก $(ก:1ก* $8!

ก1!-!* f(n) .1#ก-)*-ก*!$)ก n # +*#,

f(n) = f(n/2) + 2

2.ก!,$ก* -ก %. #&-ก*!

,$ก* a₁, a₂, ..., a_n

1.ก n = 1 +*#, a₁ 2,,1

2.<* n > 1 ,ก ก$8!8 !,1 ก, "#&ก:1 1,1 ,!,

1 ก#$ ^กก 20

# ,$ก#ก, )

ก1!-!* f(n) .1#ก-)*-ก!,1

f(n) = 2f(n/2) + 2

3.ก.* $'#ก (Merge sort) !กก 1ก* ก.

, ก2 ,#, C ก ก:1ก.* $'#ก.ก, ก, $ +* ก*#1$'#กก-!*$1 :8-)*

.1#ก1ก n

ก1!-!* M(n) .1#ก1ก-)*-ก!.* $'#ก กa₁, a₂, ..., a_n

M(n) = 2M(n/2) + n

กก 21

@>;'( $ก#ก, )

@>; -!* f 24/ก )%$:8 * ก#$$%& #ก

f(n) = a f(n/b) + c

$ n ! b # a > 1 +*#, f(n) = O(log n) <* a = 1 <* a > 1

ก.ก <* n = b^k *# f(n) = C₁n^{log_b a} + C₂ $ C₁ = f(1) + c/(a - 1) C₂ = -c/(a - 1)

1. .$.1#ก ก*#B

2. .$.1#ก% !,1 #&-* 2

fn=On^logba

กก 22

@>;$

@>; -!* f 24/ก )%$:8 * ก#$$%& #ก

f(n) = a f(n/b) + c n^d

$ n ! b # a > 1 +*#,

3. .!#$::* -ก1#'( #$$%& #ก ก.

'#ก (merge sort)

fn=

{

^{OOOnnnddloglog nba}

}

กก 23

". ก, )

1..!.1#ก!$* ก ก*#B$$)ก 64 # 2.$$#, f(n) = f(n/3) + 1 $ n ! 3 # f(1) = 1 .! f(3), f(27),

f(729)

3.$$#, f(n) = f(n/5) + 3n²$ n ! 5 # f(1) = 4 .! f(5), f(125), f(3125)

4.$$$$ n = 2^k$,%* กก * $'*)! %*$

ก$,!$ n/2 = 2^k-1ก$$'*) 2^k-1 $ - 1'*)!$

$,ก.! 2^k-2 $

.$ก#ก .1#ก,!$.ก+*$)%

!8$

กก 24

4/ก )ก, ก1

4/ก )ก, ก1ก.กก-)* ก$ก1 (power series) $*4/ก )

# x ก1!$&? ก1 # x

$ 4/ก )ก, ก1.ก1 a₀, a₁, ..., a_k, ... .1#.+$,.1ก

G(x) = a₀ + a₁x + ... + a_k x^k +...

$กก4/ก )ก, ก1.ก {a_k} "#&ก,##, 4/ก )ก, ก1$0

# , 4/ก )ก, ก1.ก1 {a_n} ก1!-!*

{a_n}, a_n = 3

{a_n}, a_n = n + 1

{a_n}, a_n = 2ⁿ

=

∑

k=0

∞

akx^k

∑

k=0

∞

3 x^k

∑

k=0

∞

k1x^k

∑

k=0

∞

2 x^k

กก 25

# ,ก-)*4/ก )ก, ก1$0

1.4/ก )ก, ก1+*.ก1 1, 1, 1, 1, 1, 1, 1

G(x) = 1 + x + x² + x³ + x⁴ + x⁵ + x⁶

,(x⁷ - 1)/(x – 1) = 1 + x + x² + x³ + x⁴ + x⁵ + x⁶

G(x) = (x⁷ - 1)/(x - 1)

2.-!* m 2.1#3$#ก a_k = C(n, k) $ k = 0, 1, 2, ..., m 4/ก ) ก, ก1$0

G(x) = C(m, 0) + C(m, 1) + ... + C(m, m)x^m

.ก@>;#B+*#, G(x) = (1 + x)^m

กก 26

#$.ก#ก ก$ก1

4/ก ) f(x) = 1/(1 - x) 24/ก )ก, ก1 1 1, 1, 1, ...

4/ก ) f(x) = 1/(1 - ax) 24/ก )ก, ก1 1 1, a, a², ...

@>; ก1! *#

ก1!-!* f(x) = 1/(1 – x)².!$&? 4/ก )ก, ก1+*.ก f 1/(1 – x) = 1 + x + x² + ...

+*#,

fxgx=

∑

k=0

∞

akb_kx^k, fxgx=

∑

k=0

∞

 ∑

^{j=0k ajbkj}



^xk

f x=

∑

k=0

∞

akx^k

1

1x²=

∑

k=0

∞

 ∑

^{j=0k 1}



^xk=

∑

^{k=0∞ k1xk}

gx=

∑

k=0

∞

bkx^k

กก 27

$$&?#$

$ -!* u 2#. k 2.1#3$+$,2 $$&?

#$ #,

# ,



^uk



⁼

{

^{uu1⋯ukk !1 1 if kif k0=0}

}



²³



^{=2343 ! =4}



¹³²



^{=121213 ! 122=161}



^nr



^=1r



^nr1r



กก 28

@>;$&?#$

$ -!* x 2#. |x| < 1 u 2.1#.

# ,.!4/ก )ก, ก1 (1 + x)^-n (1 - x)^-n$ n 2.1#3$#ก

1x^u=

∑

k=0

∞



^uk



^xk

1xⁿ=

∑

k=0

∞



^nk



^xk=

∑

k=0

∞

1^k



^nkk¹



^xk

1xⁿ=

∑

k=0

∞



^nk1k



^xk

กก 29

/0!ก4/ก )ก, ก1

$<-)*4/ก )ก, ก1-กก*/0!ก+* "(%ก.1#ก .!$,-, C

), กก.!$, r .ก n :8$$กก'(

e₁ + e₂ + ... + e_n = C

$ C 2,#, e_i2.1#3$+$,2$ +ก1!

# , ..1#'(

e₁ + e₂ + e₃ = 17

$ e₁, e₂, e₃2.1#3$+$,2 2 < e₁ < 5, 3 < e₂ < 6, 4 < e₃ < 7

.1#'(* ก ,$&?!* $ x¹⁷

(x² + x³ + x⁴ + x⁵)(x³ + x⁴ + x⁵ + x⁶)(x⁴ + x⁵ + x⁶ + x⁷)

กก 30

# ,/0!ก-)*4/ก )ก, ก1

ก$$ก /0!ก,#$ +*.ก

$ x¹⁷+*.กก,$ก x^e1ก,$ x^e2ก,$$ x^e3" e₁+e₂+e₃=17

%#,$&?!* x¹⁷ 3 :821 * ก

ก#,ก!'(ก,# .+$,+*,ก#,ก. C ,#&ก$<

1+-)*กกก#* ก4/ก )%>+*

# , ..1##&2++*!$ ก.กกก)+$,ก,ก -!*ก3ก) ก 3ก!0 3ก) ",+*กก+$,1ก#, ) +$,ก)

#&1 $-)*1!3ก,+* (x² + x³ + x⁴) $ #$3ก

$+*4/ก )ก, ก1- (x² + x³ + x⁴)³"* ก$&?

!* $ x⁸ $$ก ก2++*!ก#&.ก (3, 3, 2), (4, 2, 2)

กก 31

ก-)*4/ก )ก, ก1ก*#$$%& #ก

$<!'( #$$%& #ก+*"-)*4/ก )ก, ก1 ),

# , .!'( #$$%& #ก a_k = 3a_k-1 +$* a₀=2

#&1 -!* G(x) 24/ก )ก, ก1 1 {a_k} ก,#

ก#,

% a₀ = 2 a_k = 3a_k-1 +*#,

Gx=

∑

k=0

∞

a_kx^k

x Gx=

∑

k=0

∞

a_kx^k1=

∑

k=1

∞

a_k1x^k

Gx3 x Gx=

∑

k=0

∞

a_kx^k3

∑

k=1

∞

a_k1x^k

=a₀

∑

k=1

∞

a_k3 a_k1x^k=2

กก 32

ก-)*4/ก )ก, ก1ก*#$$%& #ก

+*#, G(x) = 2/(1 - 3x)

,

+*#,

a_k = 2 · 3^k 1

1a x=

∑

k=0

∞

a_kx^k

Gx=2

∑

k=0

∞

3_kx^k=

∑

k=0

∞

2⋅3^kx^k

กก 33

# ,ก-)*4/ก )ก, ก1ก*#$$%& #ก

# , $$! $+*ก *## n #:8* $ ,# -!* a_n

.1#! $+* n +*#,1* * ก#$$%&

#ก a_n = 8 a_n-1 + 10^n-1 ... (1)

+$* a₁ = 9 .-)*4/ก )ก, ก1! a_n

#&1 -!* (1) **# xⁿ+*

a_n xⁿ = 8 a_n-1 xⁿ + xⁿ10^n-1 ... (2)

-!* Gx=

∑

24/ก )ก, ก1 1 a₀, a₁, a₂, ...

k=0

∞

a_kx^k Gx1=

∑

n=1

∞

a_nxⁿ=

∑

n=1

∞

8 a_n1xⁿ10ⁿ¹xⁿ=8

∑

n=1

∞

a_n1xⁿ¹

∑

n=1

∞

10ⁿ¹xⁿ

=8 x

∑

n=1

∞

a_n1xⁿ¹x

∑

n=1

∞

10ⁿ¹xⁿ¹=8 x Gx x 110 x

กก 34

ก-)*4/ก )ก, ก1ก*#$$%& #ก

+*#, G(x) – 1 = 8 x G(x) + x/(1 – 10 x)

*,+*#,

+*#, a_n = 0.5 (8ⁿ + 10ⁿ)

Gx= 19 x

18 x110 x

Gx=1

2



^{18 x1 110 x1}



=1

2

 ∑

^{n=0∞ 8nxn}

∑

^{n=0∞ 10nxn}



=

∑

n=0

∞ 1

28ⁿ10ⁿxⁿ

กก 35

".

1..!4/ก )ก, ก1 1.1ก 2, 2, 2, 2, 2, 2

2..4/ก )ก, ก1 1, +-!* ,- ,,

1. 1, 2, 1, 1, 1, 1, ...

2. 1, 3, 9, 27, 81, 243, 729, ...

3. 1, 0, 1, 0, 1, 0, ...

4. -3, 3, -3, 3, -3, 3, ...

3. .!$&? $ x¹⁰$.ก ก$ก1 4/ก ), +

1. (1 + x⁵ + x¹⁰ + ...)³ 2. (x³ + x⁴ + x⁵ + ... )³

3. (x⁴ + x⁵ + x⁶)(x³ + x⁴ + x⁵ + x⁶ + x⁷)(1 + x + x² + x³ + x⁴ + ...) 4. (x² + x⁴ + x⁶ + x⁸ + ...)(x³ + x⁶ + x⁹ + ...)(x⁴ + x⁸ + x¹² + ...)

กก 36

".

4..-)*4/ก )ก, ก1% !.1##&.ก.,ก"K!$ ก 10 ก-!*3ก

4 "3ก,* +* ,* ก

5..-)*4/ก )ก, ก1% !.1##&.ก.,Lก!$!$ ก 15 # -!*3ก 6 "3ก,* +* ,* 1 #+$,$ก+ก#, 3 # 6..!.1##&.ก.," 25 )-!*1#. 4 ",+* ,*

$) ,+$,ก.3)

7. -!* G(x)24/ก )ก, ก1 1 {a_k} .! 4/ก )ก, ก1 1ก1!-!*- G(x)

1. 2a₀, 2a₁, 2a₂, 2a₃, ...

2. a₁, 2a₂, 3a₃, 4a₄, ...

กก 37

!กก%$*- ก

!กก%$*- ก<ก1+-)*-ก.1#$)ก-:.1ก : ก +*

| A ∪ B| = | A | + | B | - | A ∩ B |

1. ..1#3$#ก+$,ก 1000 !*# 7 ! 11 #

2. $$-"!,!8$ก , 1807 -$ก$ 453

#)#ก $%# 567 #) 299 #)

#ก $%# .!.1#ก+$, 3. ..1#3$#ก+$,ก 100 -!*# 5 7 +$,#

4. ..1# 8 +$,$ ก 6 #

กก 38

!กก%$*- กก:$กก#,

%.!กก%$*- ก $ -)*.1#$)ก-:.1ก$:, A, B, C +*#,

|A∪B∪C| = |A| + |B| + |C| - |A∩B| - |A∩C| - |B∩C| + |A∩B∩C|

# , $$-"!,!8$ก 1232 B> 879 B>

6 114 B>: ก.ก 103 B>6 23

B>: 14 B>6: <* ,* 2092

B>- B>!8 .!ก$B>

@>;!กก%$*- ก -!* A₁, A₂, ..., A_n2:.1ก *#

∣A₁∪A₂∪⋯∪A_n∣=

∑

1in

∣A_i∣

∑

1ijn

∣A_i∩A_j∣



∑

1ijkn

∣A_i∩A_j∩A_k∣⋯1^n1∣A₁∩A₂∩⋯∩A_n∣

กก 39

".

1. .-ก.1#$)ก ::

2. .!.1#$)ก-: |A₁ ∪ A₂ ∪ A₃| #<*$$)ก 100 - A₁, 1000 -

A₂ 10000 - A₃<*

1.A₁ ⊆ A₂ A₂ ⊆ A₃

2. ก:+$,$,#,#$กก,

3. $$)ก #!#,, : $$)ก%!8# ,-$:

3. .!.1# $!$- !กก%$*- กก: 10 : 4. ..1#3$#ก!$+$,ก 100 2.1#2ก1

.1#

กก 40

ก !กก%$*- ก

-!* A_i2:$$)ก * $ P_i.1#$)ก!$$$

P_i1, P_i2, ..., P_ik*# N(P_i1 P_i2 ... P_ik) +*#,

|A_i1 ∩ A_i2 ∩ ... ∩ A_ik | = N(P_i1 P_i2 ... P_ik)

ก1!-!*.1#$)ก+$,$$ P₁, P₂, ..., P_n N(P'₁ P'₂ ... P'_k)

-!* N .1#$)ก-: +*#,

N(P'₁ P'₂ ... P'_k) = N - |A₁ ∪A₂ ∪ ... ∪A_n |

.ก!กก%$*- ก+*

NP '₁P '₂⋯P '_n=N

∑

1in

NP_i

∑

1ijn

NP_iP_j

∑

1ijkn

NP_iP_jP_k⋯1ⁿNP₁P₂⋯P_n

กก 41

#&ก "

.!.1#'(2++*!$

x₁ + x₂ + x₃ = 11

$ x₁, x₂, x₃2.1#+$,2 x₁ < 3, x₂ < 4 x₃ < 6

#&ก " #&กก .1#% !$.1#(%+$,ก .1#3$ก1!

%.!.1#(%+$,ก 100 ก#,.1#ก $,+$,ก 100 $ 4 .1#(%+$,ก 10 .1#(%* ก#, 10 2, 3, 5, 7

.1#(%+$,ก 100 .1#!$! 2, 3, 5, 7 +$,#

-!* P₁ ก!#*# 2, P₂ ก!#*# 3, P₃ ก!#

*# 5, P₄ ก!#*# 7 $.1#(%!#, 1 <8 100

4 + N(P'₁ P'₂ P'₃ P'₄)

กก 42

ก.1#(%!#, 0 <8 100

.ก!กก%$*-!ก ก+*#,

N(P'₁ P'₂ P'₃ P'₄) = 99 - N(P₁) - N(P₂) - N(P₃) – N(P₄) + N(P₁ P₂) + N(P₁ P₃) + N(P₁ P₄) + N(P₂ P₃) + N(P₂ P₄) + N(P₃ P₄)

- N(P₁ P₂ P₃) - N(P₁ P₂ P₄) - N(P₂ P₃ P₄) + N(P₁ P₂ P₃ P₄)

N(P'₁ P'₂ P'₃ P'₄) = 99 - 100/2 - 100/3 - 100/5 - 100/7 + 100/6 + 100/10 + 100/14 + 100/15 + 100/21+ 100/35

- 100/30 - 100/42 - 100/70 - 100/105 + 100/210

– – – – – –

= 99 50 33 20 14 + 16 + 10 + 7 + 6 + 4 + 2 3 −2 −1 0 + 0

$ .1#(%* ก#, 100 21

กก 43

ก.1#4/ก )#<8

%.ก.1#4/ก )#<8.ก:$$)ก 6 #+:$$)ก 3 #

# -!* b₁, b₂ b₃2$)ก ,-: 3 #

-!* P₁, P₂, P₃$ b₁, b₂, b₃+$, ,-% 4/ก )

4/ก )24/ก )#<8 ก3, $ 4/ก )+$, * ก P₁, P₂, P₃ N(P'₁ P'₂ P'₃) = N - N(P₁) - N(P₂) - N(P₃) + N(P₁ P₂) + N(P₁ P₃) + N(P₂ P₃) - N(P₁ P₂ P₃)

$ N 2.1#4/ก )!$2++*

N(P'₁ P'₂ P'₃) = 3⁶ – C(3, 1)2⁶ + C(3, 2)1⁶ = 729 – 192 + 3 = 540

@>; -!* m n 2.1#3$#ก m > n .$.1#4/ก )#<8.ก:

$$)ก m #+:$$)ก n #!$

n^m – C(n, 1)(n-1)^m + C(n, 2)(n-2)^m - ... + (-1)^n-1C(n, n-1) 1^m

กก 44

ก.1#4/ก )#<8

.1#4/ก )#<8.ก:$ m $)ก+:$ n $)ก,ก n!S₂(m, n)

$ S₂(m, n) Stirling number of the second kind

.#&!$2++*-กก1!!*,ก-!*กก.* "

,* +* ,* !8

.#&!$2++*-ก.ก., , 6 ,ก-!*ก3ก 3

3กก* +* , ,* !8)

.!.1#'(2++* $ก x₁ + x₂ + x₃ = 13 $ x₁, x₂, x₃

2.1#+$,2* ก#, 6

.#&!$2++*-ก-,ก ก,ก -"<$"<,ก

","<* $ก ,* !8ก

..1#(%* ก#, 60