combinations and permutations
DESCRIPTION
ม.5TRANSCRIPT
-
www.clipvidva.com
1
PAT1 2 - 5 PAT1
1.
2.
3.
4.
5.
6.
-
www.clipvidva.com
2
1. (Combinatorics) (discrete mathematics : )
1.1 (Tree Diagram)
2 4 A B 2 4 0 1 5 +3 -2
-
www.clipvidva.com
3
1.2 2 - k T1, T2, T3, , Tk n1, n2, n3, , nk
n1 + n2 + n3 + ... + nk = k
ii 1
n
- k 1 n1 n2 n3 n1n2n3nk
10 8 5 3 5 A C B A B 3 B C 4 (1) (2) (3) 3 6 (1) (2) (3)
-
www.clipvidva.com
4
1 2 (1) (2) (3) 3 5 (1) (2) (3) 2 (4) - 10 (1) (2) (3) 1 9 (4) 1 0, 1 ,2 ,5 ,6, 7 (1)
-
www.clipvidva.com
5
(2) (3) 200 - 600 3 AMOUNT (1) (2) 1 (3) 5 3 3 2 1 (1) (2) 1 ( 1 ) . 1/1 4 6 (1) (2) (3) 1
-
www.clipvidva.com
6
2. ( Factorial)
n n 1 n n!
n! = 1234 (n-1) n
(1) 5! ________________________ (2) 3! + 8! ______________________
(3) 7 !
4 ! ________________________ (4)
6 !
4 ! 2 ! ______________________
(1) n!
(n 2) ! ____________________________________________
(2) (n 1) ! (n 3) !
(n 2) ! n!
____________________________________________
(1) 7654321 ____________________________________________
(2) 678 ____________________________________________
(3) (n + 3) (n + 2) ...n __________________________________
(4) (n2 - 1) (n2 - 4) (n2 - 9)n __________________________________
(5) (3n2 + 5) (3n2 + 4) (3n2 + 3) __________________________________
-
www.clipvidva.com
7
n ( n {0} ) (1) n! = 720
(2) (n 2) !
(n 1) !
= 120
(3) n!
(n 3) ! =
(n 1) !
(n 5) !
(4) (n 1) ! 8 (n 2) ! (n 1) ! = n!
149! 0
-
www.clipvidva.com
8
3. (Permutation) ( )
3.1 n r (1 r n)
Pn,r = n!
(n r) !
8 ( 1) 5 (2) 8 (3) 4
3.2 n n1 1 n2 2 k n 1 + n2 + ... + nk = n
n 1 2 k
n!
n ! n ! ... n !
n n!
-
www.clipvidva.com
9
CLIPVIDVA (1)
(2) 3
3.3 n n (n - 1)!
7 10
3 4 5
3 ()
n (n 1) !
2
n
1 2 r
(n 1) !
n ! n ! ... n !
... n1, n2, , nr 1
-
www.clipvidva.com
10
COCO 8 3 3 2 5 6 3 2 3 4 (1) (2) (3) (4) (5) (6) 2 (7) 4
-
www.clipvidva.com
11
6 6 (1)
(2)
(3) - 1
(4) - 2
(5) - 3
(6)
7 , (1) ,
(2)
(3)
(4)
(5) , 3
-
www.clipvidva.com
12
8 10 (1) 2 1 1 (2) 1 (3) 1 (4) 2 5 2, 3, 4, 5, 6 (1) (2) (3) 4 3 5 (4) 5 2 INTEGER (1)
-
www.clipvidva.com
13
(2) (3) A B
(1)
(2) P (4) P Q 6 2 (1) (2) (3) (4)
A
B
Q
P
-
www.clipvidva.com
14
7 , (1) ,
(2)
(3)
(4)
(5) , 3 - 5 (1) - (2) - 6 6 (1) - 1 (2) - 2 (3) - 3
-
www.clipvidva.com
15
5 1 14 4 2 (1) (2) (3)
-
www.clipvidva.com
16
4. (Combination)
4.1 n r (1 r n)
n,rn n!
Cr (n r) !r !
6 3 50 6/1 12 4 3 5 6
(1) 2 4
(2) 2
(3) 1
-
www.clipvidva.com
17
(4)
(5) 2 5 7 8
(1) 3
(2) 2 .. 3 5 3 3
(1) 3
(2) 3
(3) 2
-
www.clipvidva.com
18
5 52 (1)
(2) 2 3
(3) 4
(4) 5
(5)
(6) - 6 4 2 2
(1) 4 -
(2) 4 - 1
-
www.clipvidva.com
19
4 7 5 7 5x4
(1)
(2) 7 (1) (2) (3) 7
-
www.clipvidva.com
20
(4) A B 7 (4.1) A
(4.2) A B 10 10 5
(1) 10
(2)
- 5 5 2 4 5 2 3 5 8 , 6
-
www.clipvidva.com
21
4 4 3 2 (1) - (2)
4.2 n r k () r1, r2, , rk (r1 + r2 + + rk = r) n - r = R
1 2 k1 1 2
k1 2 3
n r r ... rn n r n r r...
rr r r
=
1 2 k
n!
r ! r ! ... r ! R !
9 1, 2 3 (1) 1 2 2 3 3 4 (2) 1 2 2 2 3 3
-
www.clipvidva.com
22
(3) 2 3 4 (4) 2 2 3 9 3
(1) 2 3 4 (2) 2 2 3
4.3
__________________
n 1 n + 1 ( , 1 , 2 , , n )
-
www.clipvidva.com
23
6 1 (1) (2) 1 (3) 2 5
10 3 (1) 1
Stars and Bars
(1) n r
1 n 1
r 1
(2) n r
() n r 1
r 1
(3) n r
2 n r 1
r 1
(4) n r
() n r
r
(5) n r
1 n
r
-
www.clipvidva.com
24
(2) (3) 2 (4) 3 (5) (6) 1 (x, y, z) (1) x + y + z = 10 ; x, y, z (2) x + y + z = 10 ; x, y, z {0}
(3) x + y + z 10 ; x, y, z {0}
(4) x + y + z 10 ; x, y, z
n n1 1 n2 2 k n 1 + n2 + ... + nk = n 1 (n1+1) (n2+1) (nk+1)
-
www.clipvidva.com
25
4 50 3 2 5 1 10 3 (1) (2) 1 (3) 50 1 (4) (5) 10 4 MISSISSIPPI
4 (1) 2
n
- 2n ( 2 ) 1 ( 1 ) 2n - 1
- r (r + 1)n ( r + 1 ) 1 (r + 1)n - 1
-
www.clipvidva.com
26
(2) 3 1 4 (1) 1 (2) 2 (3) 2 1 900 9 4 7 3
n r
1 r i n
i 0
r( 1) (r i)
i
-
www.clipvidva.com
27
4.4
A = {1, 2, 3} B = {a, b, c, d} (1) A B (2) A B A = {1, 2, 3} B = {a, b}
(1) A B (2) A B ENT41 .. A = {1, 2, 3} B = {a, b, c, d} { f : A B f 1-1}
A m B n
- A B ( f : A B ) nm
- A B (1 1
f : A B
) Pn,m
- A B (onto
f : A B ) n i m
i 0
n( 1) (n i)
i
-
www.clipvidva.com
28
ENT44 .. A = {1, 2, 3, 4} S = { f : A A f (x) x 1 x A }
S
4.5 Pn,r Cn,r
1. Pn,0 = 1 Pn,n = n! 2. Cn,0 = 1 Cn,n = 1 3. Pn,r = Cn,r r!
4. Cn,r = Cn,n-r 5. Cn,r + Cn,r+1 = Cn+1,r+1 (Pascals identity)
n ( n {0} ) (1) 2Pn,2 + 50 = P2n,2
(2) 14
3n 1
= 14
2n 2
Pn.r = 336 n
r
= 56 n r
-
www.clipvidva.com
29
n 1
r 1
= 120 n
r
= 165 n 1
r
4 2 6 n 1 4 n
NOTE () (The pigeonhole principle)
- k + 1 k 1 2
- n k 1
n
k
-
www.clipvidva.com
30
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
n = 0
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
5. (a + b)n
5.1
( a + b)n (Pascals triangle)
(1) (x - y)6 (2) (2x + y)4
-
www.clipvidva.com
31
5.2 (a + b)n a b n
(a + b)n = n 0 n 1 1 n 2 2 n r r 0 nn n n n n
a b a b a b ... a b ... a b0 1 2 r n
= n n r r
r 0
na b
r
Tr+1 = n r rn a b
r
T r+1 r + 1
(2x - 3y)4
(1) 3 (x + y)9
(2) 10 (x2 - 2y)13
(3) 7 11
2 1xx
n
0
, n
1
, , n
n
(a + b)n
(a + b)n 1. (a + b)n n + 1 2. (a + b)n r + 1 n - r + 1
-
www.clipvidva.com
32
(a - 3b2)10 (1) a7
(2) b12
10
2
1x
x
(1)
(2) 5
2x
(3) x (3x - 2y)5 (1) (2) 4 (3) x3
(4) (5)
-
www.clipvidva.com
33
ENT42 .. a b x-2 x4 10
42
1x
2x
a
b
ENT36 (xy - 2y-3)8 x y -4
ENT44 .. n x n
2 1x2x
9 x15 ENT32 (1 + x)43 (2r + 1) (r + 2) 3r ENT48 .. (tan x 2cot x)
8
-
www.clipvidva.com
34
ENT38 5 20 16 3 12 7 8 10 4 12 153 5 3 2 5 3 2 5 3 2 5 3 2 2 1. 67 2. 69 3. 71 4. 73
ENT31 xr 2n
2 1xx
1.
(2n) !
2n r 4n r! !
3 3
2.
(2n) !
4n r!
3
3. (2n) !
r ! (2n r) ! 4.
(2n) !
(2r) !
ENT30
1. 8
3 1x2x
x12 56
2. 8
3 1x2x
3. (x + y)n - (x - y)n x2yn-2 2n
2
n
4. (x + y)n - (x - y)n n
2 n
-
www.clipvidva.com
35
ENT41 (x + 2)n 2 3 160 320 x n 1. x = n - 2 2. 2x = n - 1 3. 2x = n + 1 4. 3x = n - 2
(x - 2y + z)8 (1) (2) x3y2z3
(3) x4y2z2
x3 (x2 + 2x - 1)4
NOTE (a + b + c)n
1. (a + b + c)n p q rn! a b cp! q ! r !
p + q + r = n
2. (a + b + c)n n 2
2
-
www.clipvidva.com
36
5.3 (a + b)n a b 1
2n = n n n n n
... ...0 1 2 r n
- (a + b)n 2n - 2n n
n n n n
... ...0 1 r n
( + 1 +
2 + + )
8 (1) 1 (2) 2 (3) 3 11 4 5 2 (1) 1 (2) 1 (3)
-
www.clipvidva.com
37
(4) 1 1 (5) 1 1 (6) 2 2 7
NOTE
-
www.clipvidva.com
38
6.
6.1 ( Sample Space) ( Event)
(1) 1 1 1 1 (2) 2 1
S E E P(E)
n(E)P(E)
n(S)
n(E) E n(S) S
-
www.clipvidva.com
39
6.2 S A, B S 1. 0 P(A) 1 2. P(S) = 1 P( ) = 0 3. P(A) = 1 - P(A') 4. P(A B) = P(A) + P(B) - P(A B) = 1 - P(A' B') P(A B) = P(A) - P(A B)
3 1 (1) 1
(2) 1
(3)
3 2 3 (1)
(2)
(3) 2
(4)
-
www.clipvidva.com
40
7 , (1) ,
(2)
(3)
(4) , 3
2 52 (1)
(2)
(3) 2
(4) 2
2 5 2 1 4
0.5, 0.3 0.2
-
www.clipvidva.com
41
A B P(A 'B) (1) P(A) = 0.3 P(B) = 0.5 P(A 'B ' ) = 0.3 (2) P(AB) = 0.6 P(AB) = 0.15 P(AB ' ) = 0.75
5 3 2 (1) (2) 2 2 4 10 0.2 3 1
3 2
B A A B P(AB) = P(A)P(B)
-
www.clipvidva.com
42
9 5 2 4 2 2 2 1 3 3
5 2 15.00 . 15 15.00 . 5 3 5
1 1
5
1
2
-
www.clipvidva.com
43
3 4 2 2
2 1
11 1 1
ENT26 2 1 6 2 ENT44 .. 10 : 1 2 1 42 - 6 3 4 12 4 1 4 2 4 3 1 () 1 4 () 1 - 2 () 1 3 1
-
www.clipvidva.com
44
ENT40 3 6 3 2 3 4 3 PAT1 ..52 8 2 6 2 48 10 3 2 1 2 ENT23 3 3 2 2 7 3 4 3 ENT35 4 4 2 2 10 9 ENT48 .. 10
-
www.clipvidva.com
45
ENT35 13 6 4 2 PAT1 ..52 4 3 2 2 ENT31 6 3 2 1 ENT46 .. 20 () 2 10 ENT44 .. 2 12 50 5 100 3 200 3 500 1 2 300
-
www.clipvidva.com
46
ENT31 4 BERBARREL ENT42 .. 10 3 5 2 49 1 2 3 2 1 51 6 4 2 1 ENT34 A 4 B 5 A B 1-1
-
www.clipvidva.com
47
PAT1 ..52 A = {1, 2, 3, 4} B = {a, b, c} S = { f f : A B }
ENT46 .. A = {1, 2} , B = {1, 2, 3, , 10}
{ f1 1
f : A B x A f(x) = x}
ENT40 1
3
8
15
4
5
ENT46 .. 3 4
2 2 2
15 4
1 1
-
www.clipvidva.com
48
PAT1 ..52 A = {(0, n) n = 1, 2, , 10} B = {(1, n) n = 1, 2, , 10}
A B 1 50 3 1 2 3
1
2,
1
3
1
4 3
A B C 1 2 3 1. P(A) P(C) 2. P(B) P(C) 3. P(A) P(B) 4. P(A) P(B) 50 4 4 4
-
www.clipvidva.com
49
.
2 5, : ,
5, 2550.
. Entrance 5. ...
Entrance Problem Book II for successful students, : Tutor Publisher, 2549.
, . + 4, : ,
...
. New Math Tests book II, Science Center, ...
. 37-48, Science Center, ...
. 3 .. 2551-,
: Science Center, ...