1. ก ก ก
1. กก ( Multiplication Principle )
() * กก +,- 1
ก ก ก ก 2 ! !
. +,- 1 "# ก $ 1n %
. +,- 2 "# ก $ 2n % ! &
( 01 +,- 1 ' $&( )* ก + ! , # $ - ' $& .
/" $& ! # - - ! ( )* . $& ! , - ! , / ก1
/" ,". - # ! , # ./" # 2 /" ก ( - ' $&
( )* $ ก %
()2,+3
1. ! - +# ' $&( )*
2. ' $&( )* ก ก ก 2 ! !
. +,- 1 "# ก . ก $ 3 %
. +,- 2 "# ก # ก $ 2 %
4 % ! , ' $&( )* - ก!( 623 =⋅ %
กก
ก - - / ก $ 1n % /" / -" % "# ก
- / ก , % - $ 2n % % "# ก ! -
- ก!( 21 nn ⋅ %

2. ( 01 +,- 2 ก"- ( ,(! - 6 ( / -" (, , " 6,5,4,3,2,1 ก ก!( -
, " " 1 ( /" (! 2 ( , ก! &7 " "!ก$ ก %
()2,+3
1. ! - +# ก (! 2 (, ก! &7 " "!ก
2. (! 2 (, ก! &7 " "!ก ก ก 2 ! !
. +,- 1 "# ก(! ( ก(! ! , 6 ( , &7 "!ก ( $ 6 %
. +,- 2 "# ก(! ก ( ก(! "# 5 ( , &7 "!ก - $ 5 %
4 (! 2 ( , ก! &7 " "!ก$ 3056 =⋅ %
( 01 +,- 3 " 4 "!ก$ ก %
1. " 1 ก! $ / -" "!ก
2. , " 1 ก! $ / -" "!ก
3. &7 +
4. &7 5 " !
()2,+3
! " !ก+: ; .ก $ , 10 ! +# 9,8,7,6,5,4,3,2,1,0
1. 67 .83 ก 946 : 1 * ก
! - +# ก " 4 "!ก " 1 ก! $ / -" "!ก
ก " 4 "!ก " 1 ก! $ / -" "!ก ก ก 4 ! !
. +,- 1 "# ก ! " ก ! " 9 ! , &7 "!ก2! $ 9 % ( ก 0)
. +,- 2 "# ก ! " ก ! " 10 ! , &7 "!ก $ 10 %
. +,- 3 "# ก ! " ก ! " 10 ! , &7 "!ก ( $ 10 %
. +,- 4 "# ก ! " , &7 "!ก - $ 10 %
4 " 4 "!ก " 1 ก! $ $ 000,91010109 =⋅⋅⋅ %
2. <67 .83 ก 946 : 1 * ก
! - +# ก " 4 "!ก , " 1 ก! $ / -" "!ก
ก " 4 "!ก , " 1 ก! $ / -" "!ก ก ก 4 ! !
. +,- 1 "# ก ! " ก ! " 9 ! , &7 "!ก2! $ 9 % ( ก 0 )
. +,- 2 "# ก ! " ก ! " "# 9 ! , &7 "!ก $ 9 %
. +,- 3 "# ก ! " ก ! " "# 8 ! , &7 "!ก ( $ 8 %
. +,- 4 "# ก ! " ก ! " "# 7 ! , &7 "!ก - $ 7 %
4 " 4 "!ก , " 1 ก! $ $ 536,47899 =⋅⋅⋅ %

3. 3. => ?3 ( ,-
+ &7 "!ก - " " + +# 9,7,5,3.1 - !
! - +# ก " 4 "!ก &7 + .$,-$ ก # $ 2, ,
/ - ! " 1 ก! $
ก " 4 "!ก &7 + ก ก 4 ! !
. +,- 1 "# ก ! " ก ! " 9 ! , &7 "!ก2! $ 9 % ( ก 0 )
. +,- 2 "# ก ! " ก ! " 10 ! , &7 "!ก $ 10 %
. +,- 3 "# ก ! " ก ! " 10 ! &7 "!ก ( $ 10 %
. +,- 4 "# ก ! " ก + 5 ! , &7 "!ก - $ 5 % ( 9,7,5,3.1 )
4 " 4 "!ก , " 1 ก! $ $ 500,4510109 =⋅⋅⋅ %
4. => ?3 ( +,- 5 (
5 " ! &7 "!ก - " " 0 /" 5 - !
ก " 4 "!ก ก ก 4 ! !
. +,- 1 "# ก ! " ก ! " 9 ! , &7 "!ก2! $ 9 % ( ก 0 )
. +,- 2 "# ก ! " ก ! " 10 ! , &7 "!ก $ 10 %
. +,- 3 "# ก ! " ก ! " 10 ! &7 "!ก ( $ 10 %
. +,- 4 "# ก ! " ก ! " 2 ! , &7 "!ก - $ 2 % ( 5,0 )
4 " 4 "!ก , " 1 ก! $ $ 800,1210109 =⋅⋅⋅ %

4. 2. กก (ก ( Multiplication Principle )
( 01 +,- 1 ($2- 1 ( ก$2- !( , 52 ( % ($ $2- 2 # 2/
# " , !
()2,+3 ก ($2- 1 ( $ 2 # 2/ # " , ! , / ก &7 3 ก : !
ก ,+,- 1 "# ก ($2- 2 1 ( ก$2- 2 ! , 13 ( $ 13 %
ก ,+,- 2 "# ก ($2- 2/ 1 ( ก$2- 2 ! , 13 ( $ 13 %
ก ,+,- 3 "# ก ($2- " , ! 1 ( ก$2- 2 ! , 13 ( $ 13 %
4 ($2- 1 ( $ $2- 2 # 2/ # " , ! $ 39131313 =++ %
( 01 +,- 2 ,( ก "!ก &7 +-/" / -" "!ก " $,-1 ก! , ! , ก
()2,+3 /(- ! , ก ก &7 2 ก"B-, !
ก A1<+,- 1 & ก ( "!ก - &7 " 0
"!ก - "# ก$ 1 % +# " 0 - !
# ก " Bก "!ก &7 " $,-1 ก! C ! $ -
"!ก ( "# ก$ 9 % ก " 9,,2,1 K
"!ก "# ก$ 8 % ก " "# 8 !
"!ก2! "# ก$ 7 % ก " "# 7 !
4 ก"B-, , ! , 5047891 =×××
ก A1<+,- 2 "!ก - &7 " 6,4,2 # 8
"!ก - "# ก$ 4 % ก " 6,4,2 # 8
# ก ! " 0 ! $,- ก "# ก C ! "# ก " "!ก2! ก- /" "# ก "
"!ก # D "# ," !(
"!ก2! "# ก$ 8 % ก " "# 1 $,- - " 0 /" " "!ก -
กก (ก
ก , %ก k % +# % 1 % k
ก % 1 , % 1n %
ก % 2 , % 2n %
M M
ก % k , % kn %
/" %ก / -" %/ ก - ก! /" % - ก!( k21 nnn +++ K %

5. "!ก "# ก$ 8 % ก " 1 $,- - " "!ก - /" " "!ก2!
"!ก ( "# ก$ 7 % ก " "# 7 !
4 ก"B-, , ! , 792,17884 =×××
B* ,( ก "!ก &7 +-/" / -" "!ก, " $,-1 ก!
, ! , 296,2792,1504 =+
( 01 +,- 3 2 !E ก, ,+, ก ! 2 "!ก # 3 "!ก # 4 "!กก$
/ - , 2 !E 1 ก! / -" "!ก /" ! $ ก/((
()2,+3 ก ,+,- 1 ! & ก ($& 2 !E 2 !
. +,- 1 "# ก2 !E "!ก 1 $ 4 %
. +,- 2 "# ก2 !E "!ก 2 $ 3 %
/ - ! 2 !E 2 ! $ 1234 =⋅ /((
ก ,+,- 2 ! & ก ($& 2 !E 3 !
. +,- 1 "# ก2 !E "!ก 1 $ 4 %
. +,- 2 "# ก2 !E "!ก 2 $ 3 %
. +,- 3 "# ก2 !E "!ก 2 $ 2 %
/ - ! 2 !E 2 ! $ 24234 =⋅⋅ /((
ก ,+,- 3 ! & ก ($& 2 !E 3 !
. +,- 1 "# ก2 !E "!ก 1 $ 4 %
. +,- 2 "# ก2 !E "!ก 2 $ 3 %
. +,- 3 "# ก2 !E "!ก 2 $ 2 %
. +,- 4 "# ก2 !E "!ก 2 $ 1 %
/ - ! 2 !E 2 ! $ 241234 =⋅⋅⋅ /((
4 ! $ ! , - ก!( 60242412 =++ /((
:Cก+ ,0 ( factorial )
! +# n)1n(321!n ×−××××= K
# 123)2n()1n(n!n ××××−×−×= K
!E"!กF:. G !n H - - G /)ก " H # G /)ก " H
+ )0 < n &7 ,( ก /)ก " n +# I"+: ,( ก ! / - 1 n
/" / !n

6. ( 01 +,- 1 1!1 =
212!2 =×=
6123!3 =××=
241234!4 =×××=
12012345!5 =××××=
( 01 +,- 2 +-
1.
!678
!61234
!8
!6!4
××
××××
=
⋅
7
3
=
2.
!5
!5678
!5
!8 ×××
=
678 ××=
336=
()2, ,0 E = ,-0 ( permutation )
1. % !( &" ( Linear Permutation )
% !( &" &7 ก ! / 1 /(- ก$ &7 2 /((
!( ก , ! +#
% !( &" / ก - ก! ! ,
( 01 +,- 1 , ! # / ก - ก! 6 "-, ก , 4 "-, 2# ! &7 / ( ! ! $ ก %
()2,+3 ก
)!rn(
!n
p r,n
−
=
6n = , 4r =
)!46(
!6
P 4,6
−
=
!2
!6
=
3456 ×××=
360=
4 ! ! # 4 "-, $ 360 %
% !( &" n 1 / ก - ก! ! ,
! + " r )nr1( ≤≤ - ก!( r.nP %
)!rn(
!n
p r,n
−
=

7. ( 01 +,- 2 n &7 ,( ก /" +- n 1 70
)!2n(!3
!n
=
−⋅
()2,+3 70
)!2n(!3
!n
=
−⋅
!370
)!2n(
)!2n()1n(n
⋅=
−
−⋅−⋅
2370)1n(n ⋅⋅=−
420)1n(n =−
1.2 % !( &" $,-/ ก - ก! ! ,
( 01 +,- 3 % ! ! !กF ! , ก+ - GCHAKKARINH 1 ! !กF A $,- - ก!
- ก!( -
()2,+3 ! !กF ! , 9 !
, ! !กF A - 2 !
, ! !กF K - 2 !
/" , ! !กF C, H , R , I /" N - " 1 !
% !( &" ! !กF $,-, # $ - ก!(
!2!2
!9
!1!1!1!1!1!2!2
!9
×
=
××××××
%
! A - ก! , % ! $ - ก!(
!2
!8
%
4 % ! !กF A $,- - ก! - ก!( 560,70
!2!2
!87
!2
!8
!2!2
!9
=
×
×
=−
×
%
2. % !( &" ก", ( Circular Permutation )
( 01 +,- 4 ! !ก 10 + ! ( ] ก", 1 , 10 ! $ ! , ก %
()2,+3 % ! !ก 10 + ! ( ] ก",1 , 10 ! - ก!(
!9)!110( =−
880,362= %
- n , 1n ,# ก! &7 ก"B-, , 2n ,# ก! &7
ก"B-, … , kn ,# ก! &7 ก"B-, k nn...nnn k321 =++++
% !( &" ก"B-, n - ก!(
!n!n!n
!n
k21 ××× K
% !( &" ก", / ก - ก! n - ก!( )!1n( − %

8. ( 01 +,- 5 , !ก 6 + /" !ก E 6 + ก ! + ! , ! ( ] ก",
1 , 12 ! !ก ก!( !ก E ! "!(ก! , % ! ! , ก %
()2,+3 ! !ก 6 + ! ( ] ก",ก- !ก 2 + D
$,- ! ก ก! $ !5 %
/ -" % ! !ก E ! ก - 1 , 6 ! $ !6 %
4 % ! !ก /" E ! "!(ก! - ก!( 400,86!6!5 =× %
( 01 +,- 6 , 5 + /" E 4 + ก ! + ! 9 + # &7 ก", $,-, E + "
# ก! , % ! ! , ก %
()2,+3 %ก ! ก + ! 9 + # &7 ก", $,-, E + " # ก! ก+# ! #
&7 ก",ก- "! ก ! E # / ก - ! &
# กก ! # &7 ก",ก- ! $ !4 %
/" ! / - E $& # / ก, ! , 5 / - /" / -" % , ! I E
$& # $ !4 % ! E ! $ !45× # !5 %
4 % ! + ! 9 + # &7 ก", $,-, E + " # ก!
$ 880,2!5!4 =× %

9. ()2,ก ?4 <1 ( Combination )
( 01 +,- 1 % ! , "# ก ( " ก 3 ก " ก ! , 12
()2,+3 % ! , = 3
12
C %
=
!3)!312(
!12
×−
=
123
101112
⋅⋅
⋅⋅
= 220
4 % ! , - ก!( 220 %
( 01 +,- 2 ,& กก - 12 / - 1 / -" / - , - ก! ก (+ ! " 5 / -
/ -" + ! ( ,& กก - , %ก ( ! ,
()2,+3 ก ./ -" + ! ( ,& กก - , / - , "# ก ( # D
$ ก 2 4 / - ก& กก ! , 11 / - ( $,- ,ก!(& กก )
% ( ! , = 4
11
C %
=
!4)!411(
!11
×−
=
1234
891011
⋅⋅⋅
⋅⋅⋅
= 330
4 % ! , - ก!( 330 %
( 01 +,- 3 ", ! # 7 "-, )* , ! # 9 "-, ก ! ก /"ก ! # ก!
+ " 2 "-, , %/"ก ! # ! ก %
()2,+3 %/"ก ! # , ! !
. +,- 1 " "# ก ! # ! 2 "-, ก , - 7 "-, 2# /"กก!()*
$ 21C2
7
= %
. +,- 2 )* "# ก ! # ! 2 "-, ก , - 9 "-, 2# /"กก!()*
$ 36C2
9
= %
4 ก ! + , %/"ก ! # ! - ก!( 7563621 =⋅ %
% ! ,- / ก - ก! n "# ก+ " r )nr0( ≤≤
- ก!(
!r)!rn(
!n
−
%
!E"!กF:./
!r)!rn(
!n
r
n
CC r.nr
n
⋅−
=





==