4 1.. 3. 4. 1. 1 3 4 5 6 1 3 4 5 6. 1 1 1 A B P (A B) = P (A) + P (B) P (C) = P (A) P (B) 3. 1 1 P (A) = 1 P (A) A A 4. A B P A (B) = n(a B) n(a) = P (A B) P (A) 50
015 016 018 1 4 5 8 8 3 T 1 3 1 T T 5 1 5 T 1 5 0 A 0 1 A 1 A P (A 1 ) = C 1 6 C 8C 3 = 15 8, P (A ) = C 6 C 1 8C 3 = 3 8 T 5 B P (A 1 B) = 15 8 1 3 = 5 8, P (A B) = 3 8 3 = 1 14 P (A B) P (A 1 B) + P (A B) = 1 14 5 8 + 1 14 = 5 + = 7 n! 1. n P r = (n r)! = n(n 1)(n ) (n r + 1), np 0 = 1, 0! = 1. n P n = (n 1)! n 3. n p q r n! p + q + r + = n p!q!r! 4. n C r = n P r n! = r! r!(n r)!, nc n = 1, n C 0 = 1, n C r = n C n r 5. n C r = n 1 C r 1 + n 1 C r 6. 1 A p n A r n C r p r (1 p) n r 3 n r 1 n 1 C r 1 1 n 1 C r 011 5 {p + (1 p)} n 51
A P (A) A A A, B A B A 4 B 7 C 9 (1) A, B, C P (A) =, P (B) =, P (C) = () C A A C (3), 0 P (A B) P (A)P (B) P (A C) P (A)P (C) 0 < 1 = > (4) 1 A B A C A, B, C 1 018 5
4 A, B, C 3 1 1 (1) A, B E 1 (),, 0 5 A, B, C 3 E 3,, 0 A 1 A B 3 B 4 C 5 C (3) E 1 E (4),, 0 5 B, C E 3,, 0 A 1 A B 3 B 4 C 5 C A, C E 3 (5) 0 6 E 1 E p 1, E E p, E 3 E p 3 0 p 1 < p < p 3 1 p 1 > p > p 3 p 1 < p = p 3 3 p 1 > p = p 3 4 p 1 = p < p 3 5 p 1 = p > p 3 6 p 1 = p = p 3 017 53
4 3 5 1 1 A 1 B 1 (1) A B 1 () A B A B (3) A 1 B A A B A B A B B 016 54
018 B C 6 6 = 36 a, b (a, b) { } (1) A = (4, 1), (4, ), (4, 3), (4, 4), (4, 5), (4, 6) 6 P (A) = 6 36 = 1 6 { } B = (1, 6), (, 5), (3, 4), (4, 3), (5, ), (6, 1) 6 P (B) = 6 36 = 1 6 { } C = (3, 6), (4, 5), (5, 4), (6, 3) 4 P (C) = 4 36 = 1 9 () C { } C = (3, 6), (4, 5), (5, 4), (6, 3) 4 { } 4 A C A = (4, 5) 1 { } A 6 C C A = (4, 5) 1 P C (A) = P A (C) = n(c A) n(c) n(a C) n(a) = 1 4 = 1 6 (3) 1 1 { } A B = (4, 3) 1 P (A B) = 1 36 P (A)P (B) = 1 6 1 6 = 1 36 P (A B) = P (A)P (B) 1 P (A C) = 1 36 P (A)P (C) = 1 6 1 9 = 1 54 P (A C) > P (A)P (C) (4) 1 { } A C = (3, 6), (5, 4), (6, 3) 3 P (A B)P (A C) = 1 36 3 36 = 1 43 B C B C = BB C = C B C B C A, B, C 1 55
4 1 ( i ) A C A B 1 A C B ( ii ) A B A C 1 B A C (iii) A B A C 1 A B C (iv) A C A B 1 C A B { } A B = (1, 6), (, 5), (3, 4), (5, ), (6, 1) 5 ( i ), ( ii ) 5 36 1 36 (iii), (iv) 3 36 ( 5 36 + 3 ) 36 = 1 81 017 ABC 6 A B C ( i ) 5 ( ii ) 3 (iii) 1 (iv) ( v ) (vi) 4 1 3 = 4 4 = 1 6 4 3 1 = 4 4 = 1 6 4 3 1 = 4 4 = 1 6 4 3 1 = 4 4 = 1 6 4 3 1 = 4 4 = 1 6 4 1 3 = 4 4 = 1 6 0 (iii)( v )(vi) ( ii )(iv)(vi) 4 ( i )(iv)( v ) (1) A B (vi) E 1 = 1 1 6 = 5 6 () E 1 3 5 E = 1 6 + 1 6 + 1 6 = 1 (3) E 1 ( i )( ii )(iii)(iv)( v ) 5 E ( i )( ii )(iii) 3 3 5 56
(4) E B C (iv) E 5 (iv) 4 0 ( i ) ( ii ) 5 3 0 3 5 E 5 1 6 5 = 5 6 E 3 1 6 5 = 5 6 (5) 1 1 (a) (3) p 1 = 3 5 (b) (4) E 5 3 p = 3 5 (c) (4) E 3 5 3 p 3 = 3 5 (a)(b)(c) p 1 = p = p 3 6 016 n C r 1 1 11 A B (a) 4 3 = 1 (b) 4 3 = 1 (c) 4 5 = 0 A B (d) 3 4 = 1 (e) 3 = 6 (f) 3 5 = 15 A B (g) 5 4 = 0 (h) 5 3 = 15 (i) 5 4 = 0 (1) (i) 0 1 0 1 11 = 1 5 33 = 8 33 () A B (c) 0 0 1 11 = 5 33 A (a)(b)(c) 44 B (c) 0 0 44 = 5 11 (3) B (c)(f)(i) 55 A (i) 0 0 55 = 4 11 57
A B (c) 0 0 1 11 = 5 33 A B (f) 15 15 1 11 = 5 44 A B (i) 0 0 1 11 = 5 33 B 5 33 + 5 44 + 5 33 = 0 + 15 + 0 1 11 = 55 1 11 = 5 1 5/33 5/1 = 1 33 = 4 11 017 016 58
3 011 011 1 4 p 5 q 1 8 (1) 8 4 3 p 3 q 5 1 4 7 4 p 3 q 5 1 5 7 4 3 p 3 q 5 () 0 7 0 7 C 7 C 3 1 8 C 1 8 C 7 C + 7 C 3 3 8 C 1 + 8 C 4 7 C 4 7 C 5 5 8 C 6 8 C 7 6 7 C 4 + 7 C 5 7 8 C 6 + 8 C 7 (3) 8 4 3 n = 1,, 3, 4, 5, 6 n 4 n 4 3 0 6 p p q q 3 4 p = 4 6 = 3 5 q = 6 = 1 3 (1) (p + q) 8 p 3 p 3 q 5 8C 3 p 3 q 5 = 56p 3 q 5 q 7 C p q 5 = 7 C p 3 q 5 = 1p 3 q 5 q 7 C 3 p 3 q 4 = 7 C 3 p 3 q 5 = 35p 3 q 5 () 3 n C r = n 1 C r 1 + n 1 C r 8 C 3 = 7 C + 7 C 3 = 7 C 5 + 7 C 4, 6 (1) 8 C 3 = 7 C + 7 C 3 (3) 6 qqqqqppp 1 p 3 q 5 3 5 3 4 5 4 5 3 q p 5 C p q 3 = 5 C p 3 q 5 = 10p 3 q 5 59
1 9 (1) A, B, C 3 () 3 3 (3) 5 3 (1) A 3 9 C 3 B 3 6 C 3 C 3 3 C 3 9 C 3 6 C 3 3 C 3 = 84 0 1 = 1680 () 9 C 3 6 C 3 3 C 3 = 80 3! (3)! 9 C 7 C 5 C 5 = 378! (1) () (1) 4 ABCD 4 P 4 = 4 1 4 C 1 3 C 1 3 C 1 4 1 C 1 4 C 1 3 C 1 C 1 1 C 1 = 4 3 1 = 4 n P r n C r 4 (1) () (1) 1 5 5! 5 = 4!! 4!! = 48 () 4 1 4 4! 1 4! = 4 4 abcd b a c b d c a d c d d a a b b c 60
3 5 6 (1) P Q () P R Q (3) P R Q P R Q 5 4 (1) 9! = 9 8 7 6 = 16 5! 4! 4 3 1 () P R 4! = 6 R Q 5! = 10!! 3!! 6 10 = 60 (3) (1)() 16 60 = 66 4 () P x 1, y 1 3 P (1) 3 P (, 1) () 3 P (, 1) (1, 1) 3 y ( b 3 (1) 1 + 1 ) 3 ( ) 3 C 1 1 1 = 3 3ab 8 () (, 1) 3 (1, 1) 3a b 3 a 3 O x 5 x + y + z = 10 xyz 10 5 3 xyz (x, y, z) = (5,, 3) 10 1! = 66 10!! nr n+r 1 C r = n H r 61