B4C2

- ( )( ) B=A A A k A A A k (B)=(A )+(A )+ +(A k ) (B) B A A A k B (Patitios) Ex. 6 4 As. ()(A )=(U) (A) ()(A B )=((A B) )=(U) (A B) (DeMoga). (A-B)=(A) (A B) Ex. A={x x N x 0 6 } B={x x=0k k Z} (A B)= As 900 ()(A B)=(A)+(B) (A B) ()(A B C)=(A)+(B)+(C) (A B) (B C) (C A)+(A B C) ()(A A A t )= t i= ( A ) ( A A ) + ( A A A ) +( ) i i j i j i i, j, k j k t+ (A A A t ). Ex. 000 000000 As 997948 Ex4. 5 6 60 As Ex5. 60 4 9 4 4 0 () () () As 9 9 Ex6. () As 5 7 () As 4 7 4---/9

. p q p q ( p )(- q ) p q ( p )(- q ) [] Ex7. 4 4 As 0 Ex8. 00 5 As 4 ( )( ). k N= k ( ) Ex9. 4 As 4 Ex0. As 6 Ex. (a+b+c+d)(e+f+g)(x+y+z+u+v) As 60 Ex. () 0 50 00 As 95 () 6 0 50 4 00 As 5. Ex. ABCD A B As 5 Ex4. A B A B B As 0 Ex5. 9 6 As 9 4---/9

. () A B (A)= (B)= A B A B={(a b) a A b B} (A B)=(A) (B) ()A B C={(a b c) a A b B c C} (A B C)=(A)(B)(C) ()A ={(a a a ) a i AA i= ------ } (A )=[(A)] Ex6. A={ } B={a b} A B= As {( a) ( b) ( a) ( b) ( a) ( b)}. Ex7. As 0 Ex8. As 5 4 Ex9. 5 As 5 4 4 4. Ex0. 5 A B A B As 6 Ex. A B Q Q As 00 5 B A Ex. A B () () P As 9 5 P B Ex. As 50 A Ex4. 999 0 0As 80 89 Ex5. a=67500 ()a ()a 9 5 As 60 5 4---/9

Ex6. 9 As 8 Ex7. 4 5 () () As 60 Ex8. As 60 Ex9. 4 As 84 Ex0. As 4 Ex. 8 8 As 56 06 88 Ex. As 0 Ex. ** ### ** AA ZZ ### 00 999 AA 00 AA 00 AA 999 AB 00 AB 00 AB 999 AC 00 AZ 999 BA 00 BB 00 PJ 888 QA 666 As 676 Ex4. As 90 Ex5. 00000 999999 999999 456 As 54 4---4/9

Ex6. A B P ()A B ()A B P ()A B Q Q (4)A B P Q (5)A B P (6)A B Q (7)A B P Q (8)A B P Q As 40 05 7 0 5 68 9 4 A B Ex7. A B B () A B C D () A C B () A C D B A (4) A C D B As 56 80 40 88 Ex8. 00 000 5 As 4 Ex9. 500 () 7 () 7 As 57 86 Ex40. 00 () ()5 ()7 5 As 50 40 80 Ex4. () 0 0 () 0 7 As 6 Ex4. 500 000 5 7 5 7 As 7 0 Ex4. As 55 Ex44. 00 () 4 6 () 4 6 As 67 4---5/9

Ex45. 000 ()4 6 9 () 4 6 9 () 4 6 9 As 89 78 97 Ex46. 800 4 4 5 4 5 As 400 480 0 Ex47. 50 5 0 As 6 Ex48. 000 500 00 50 00 As 5 Ex49. 5 4 0 5 () () As 9 99 Ex50. 5 0 0 50 As 6 Ex5. 985 () 0 () 985 0 As 596 865 Ex5. "6" As 700 Ex5. 5400 x 5400 (x 99)= x As 48 8600 8 00 8 Ex54. 60 A B C (A)=4 (B)=6 (C)=7 (A B C)=0 () (A B C )(A A ) () ((A B) (B C) (C A)) () (A B)=6 C A B As 50 5 8 Ex55. 00 As 99 4---6/9

Ex56. 8 5 0 4 6 As Ex57. 50 5 40 45 () () As 5 5 5 0 Ex58. 60 A B C A 4 B 6 C 7 8 () () As 9 Ex59. As 0 Ex60. 00 00 00 00 () () () As 4 Ex6. x y z 7 As 48 Ex6. T={ 4 5 6} S={(A B) A B T} (S)= As 6 Ex6. 8 a a + =pa + +qa + +a (p q )= 8 As ( ) 8 4---7/9

- (Pemutatio). () m P m! () () P = ( )( ) ( m+ ) = m ( m)! ()! P = () 0!= (4)!=... ( N) 6 8 9 Ex64. As 0 6 04 P P P 4 + Ex65. () N 0 P P = 4P P = As 0 0 0 () N P = P + 4 0 6 P = As 90 Ex66. 5 () As 96 () As 78 () As 64 (4) As 5 (5) As 44. () k m m ------ k m k ( m +m +...+m k =)! m! m! m! () ( ) ( ( )... 4. Ex67. A B B () () P Q Q () P Q As 0 6 0 P k A 4---8/9

Ex68. A B As 8 B A Ex69. A(-4 -) B( ) () () () As 50 8 4 Ex70. 5 5 As 8800 Ex7. () () () As 6 0 80 Ex7. 4 () () () As 4 8 ( ) Ex7. As 600 Ex74. () () () (4) (5) (6) As 840 0 40 70 480 600 Ex75. factoig () () () As 60480 504 84 Ex76. abcde ()cdbae () 70 As 6 cebda ( ) 4---9/9

Ex77. 0 4 5 () As 00 () As 640 () As 40 (4)5 As 6 (5)4 As 4 (6) As 70 (7) 0 As Ex78. 0 As 60 50. m m Ex79. 5 As 5 Ex80. 6 4 As 4096 Ex8. 5 ()4 ()5 ()6 (4)7 As 8 4 76 4 Ex8. 5 4 As 90 ( ) + + P!. = = ( )!. m (m ) = = P m! = m m ( m)!. ( ). ( ) Ex8. ----- 8 5 () () As 44 840 Ex84. 4 4 As 680 4---0/9

Ex85. ----- 8 () () () (4) (5) (6) As 5040 440 960 9 40 440. k k m ( =km) = = Ex86. P 0 As 7! 8 4 k k ()8 ()0 8 P 4. a b ( =a+b) = = P Ex87. 0 As 5 9! Ex88. () 8 () 8 () 6 As 0080 060 40 5. ( )! ( )! = Ex89. 5 () () () As 6 4. () () () (4). (k ) ( k )( ) + ( k ) a + a = k( k ) Ex90. 9 6 P 9 6 As 6 4---/9

Ex9. 6 As 60 44 80. () () Ex9. 0 () ( ) () ( ) () (4) ( ) (5) (6) (7) ( ) As 7560 0080 60 8900 600 40 7800 Ex9. () N P P + = 5 = As 7 8 8 () N P = P = As 8 + () N P = 0P = As 4o5 0 8 9 Ex94. P = 0 a a (A) P (B) P (C) P 6 5 5 5 0 (D) P 4 0 As B + Ex95. P 4 4P = 0P As 5 Ex96. a b c d e 4 5 ()(a-)(b-) 0 ()(a-)(b-)(c-) 0 ()(a-)(b-)(c-)(d-4)(e-5) 0As As 78 64 44 Ex97. 5 5 5 5 4 As 45 Ex98. 6 6 () As 64 () As 65 4---/9

Ex99. A B B As 0 A Ex00. () B A B As () A B As 0 A Ex0. 7 () As 40 () As 70 () As 440 (4) As 960 (5) As 880 (6) As 600 (7) As 40 (8) As 440 (9) As 44 (0) As 70 () As 6 () As 960 Ex0. 0 5 As 040 50 Ex0. () () () As 504 68 9 Ex04. 4 As 88 576 Ex05. 4 5 As 880 Ex06. 7 As 40 Ex07. As 5 4---/9

Ex08. As 0 Ex09. As 4 Ex0. () () As 840 40 Ex. abscissa () () s ()a s (4)b c i (5) a s (6)b c As 60 60 60 560 960 50 Ex. ()baaa ()mississippi As 60 4650 Ex. Pallmall 8 () () l () l (4) l (5) l (6)m a (7) As 840 60 780 60 4 90 54 Ex4. a b c d e As 60 Ex5. aabbcc As 0 Ex6. a b c d e () ()a e ()a e (4)a e As 0 48 7 Ex7. 0 4 5 6 7 8 9 () ()5 () (4) As 648 6 8 55680 Ex8. 0 4 5 6 7 ()4 () As 70 498 Ex9. 0 4 As 940 500 4---4/9

Ex0. 4 5 ()45 () 40 As 55 45 Ex. A B C D E As Ex. 0 4 As 00 Ex. 0 4 5 As 80 58950 60 Ex4. 4 5 6 ( ) As 50 Ex5. 5 () As 4 () As () As (4) As 80 (5) As 80 (6) As 50 Ex6. ----- 8 As 400 Ex7. 8 () () () (4) As 60 50 70 840 Ex8. 8 () () () (4) (5) (6) (7) (8) (9) (0) () () As 5040 440 70 44 96 48 0 576 4---5/9

Ex9. 6 () () () (4) As 60 4 7 48 Ex0. () () () As 0080 40 700 Ex. ----- 0 ( ) () () As 75600 9070 Ex. 9 ( ) 9! As Ex. 7 5 As 40 Ex4. () 6 () () As 0 90 80 Ex5. 0 5 0 As 5 Ex6. 0 0 As 67 Ex7. 5 ( 5 ) As 7 Ex8. 4 As 88 4---6/9

Ex9. () 5 () 8 As 0 560 Ex40. 4 4 As 90 6 Ex4. () () 6 () As 89 5 65 Ex4. ( 4 5 6)5 a b c d e (a b)(b c)(c d)(d e)=0 As 406 4---7/9

- (Combiatio). ( ) P! ( )( ) ( + ) C = = = ( N Z 0 )!!( )! ( )( ) ( ) ( vs. ) Ex4. As 4o C 8 C 9 C 0 Ex44. () P + 4 = 84C () C = C As 6o7 Ex45. k >k C = = As 4 k C k C k +. () C = ( N )( ) ( m -m ) () C = C x=y x+y= x () C = C = 0 (4) P m C y = C m! (5) C + C =C ( N )( + = ) Ex46. P : Pk : 7 C k = C 4 ( k) As (0 4) k = Ex47. () C = C 8 47 47 C 7 = () C = C = As 40 5o4 + 5 4 98 99 Ex48. () C + C + C + + C + C = Cm 0 86 C m 4 5 6 89 () C + C + C + C + + C = m<5 ( m) As (00 ) m<0 ( m) As (90 4). () C () C m () m C ( C ) (4) m C C C m m C m (5) (6) C m (7) m C C + Ex49. 5 () () As 0 0 4---8/9

Ex50. A B C D A E F G H ( )As 5 06 Ex5. 4 5 () () As 5 6 Ex5. () () As 60 0 Ex5. () () As 64 40 Ex54. 9 9 6 () As 680=C CC 9 6 () As 80= CCC! 9 6 () As 680= CCC (4) 4 As 60=C C C (5) 4 As 60=C C 9 5 (6) 4 As 7560=C C C! (7) 4 4 As 60=C C 9 5 4 9 5 4 9 5 (8) 4 4 As 5=C C C! 9 5 (9) 4 As 890= CCC!! C 4 C 9 5 4 4 4 4 4 4 Ex55. 8 () As 5796 () As 656 () As 68 (4) As 568. m ( ). x +x +...+x =m(m N). H m = C + m m H m H m ( )( ) 4---9/9

Ex56. 5 () As P 5 () As 5 5 () As C 5 (4) As H Ex57. ()x+y+z=0 As ()x+y+z=0 As 7 ()x+y+z=0 As 66 (4)x+y+z+u 9 As 6 (5)x+y+z+u<9 As 70 Ex58. x+y+z+u =0 x y z u N As 06 Ex59. xyzu=88 x y z u N As 560 Ex60. abc a b c 9 () a<b<c As 84=C 0 () a>b>c As 0=C 9 () a b c As 65= H (4) a b c As 9= H 0 Ex6. a a a b b c c d 8 4 4 As 6 6 Ex6. 4 () 4 () As () ()896=( 4 -)*4! Ex6. 7 () () 7 6 As 5(=C ) 0(= C ) 4 Ex64. 5 6 As C 5 4---0/9

Ex65. As 5 Ex66. 000 0 As 6 Ex67. { 4 5 6 7 8 9} 5 ( ) 5 As 495 Ex68. 4 ------ 0 () () () 4 As 450 00 795 Ex69. 5 7 9 4 6 As 70 Ex70. () As 84 () As 0 Ex7. () () () (4) As 80 45 5 57 Ex7. 6 5 4 As 50 Ex7. 78 As Ex74. 5 4 () () () As 0 0 80 Ex75. 5 () () As 90 0 Ex76. A B C A B C As 7 4---/9

Ex77. 6 4 0 5 As 400 Ex78. As 50 Ex79. 9 As 0 Ex80. 0 A B C A 4 B C As 400 0 Ex8. 7 () () As 49 84 Ex8. As 490 Ex8. 5 4 () () () As 5 85 8 Ex84. () 5 () As 56 Ex85. 6 () () As 540 5 Ex86. 4 6 () As 5 () As () As 40 (4) As 8 Ex87. 8 As 5796 () () As 656 () As 68 (4) As 568 4---/9

Ex88. 0 () As 6 () As () As 5 Ex89. 0 As 0 56 Ex90. 5 5 ()5 ()TwoPais ()FullHouse As 548 55 744 Ex9. 0 () () () (4) 5 (5) 5 As 86 65 0 80 76 Ex9. 5 ( 5 ) () As 4 () As () As 5 (4) As 4 Ex9. 6 6 () () As 79 540 Ex94. 7 () 4 As 05 () As 05 Ex95. 7 5 As 05 P y C y x x Ex96. =60 =0 (x y)= As (5 ) 4 Ex97. () C 0 + C + C + C + + C = 90 = 4 0 8 9 0 0 () x= C + C + C + + y= + C + C + + x+y= As 9 C C C 0 C 4---/9

k m Ex98. C 7 = C0 C 4 = C m= As 8 k= 4 Ex99. 0 () () As 9 056 Ex00. ABCD EFGH As 0 Ex0. 6 4 As 7 9 Ex0. 04 As 5 Ex0. As 6 Ex04. As C 4 Ex05. () () As 60 40 5 Ex06. a b c d e f () () () (4) As 60 60 5 90 Ex07. 7 () 4 () As 05 05 Ex08. 0 A B C ()A4 B4 C () 4 As 50 9450 Ex09. As 50 Ex0. 9 As 0 4---4/9

Ex. 0 A B C A 4 B C As 400 0 Ex. a b c d e f a b As 90 65 Ex. 6 () () 4 () 5 (4) 6 As 540 560 800 70 Ex4. 8 As 87= H 6 8 Ex5. 4 () () As 65 70 Ex6. 5 0 As 66 Ex7. x+y+z+u 0 () () As 00 0 Ex8. x+y+z+u=8 () () () (4) (5)x> 5 y> 4 z>6 u As 0 680 0 56 969 4 Ex9. () ( ) As H () As H Ex0. 4 6 5 As 7 Ex. 45 () () 5000 () 400 As 70 4 58 4---5/9

-4 N x y R. ( x + y) = C x + C x y + C x y + + C x y + + C 0 C x y. ( + x ) = C + C x + C x + + C x + + C 0 Ex. (x y) 7 x 4 y 5 As 6048 x y = = 0 C x y Ex. (9 x + ) x 7 7 () ()x As 495 Ex4. (0.98) As 0.785 Ex5. As 9 0 Ex6. ( x ) ( x x + ) As 0x -0x+ Ex7. ( x)+( x) + +( x) x As ( 6 ) Ex8. (x+)(x+)(x+)...(x+0) x 8 As 0 0 + = C 0 C + C + ( ) C = 0. x= y= C C + C + + C + C 4. x= y= Ex9. 0 0 0 C + C + C + C 0 0 = As 0 Ex0. C C C + + 4 + + C =80 As 8(HINT x= y=) Ex. C + C + C + + C = (HINT ) 5.C 0 + C + C 4 + = C + C + C 5 + = 5 0 Ex. C = C = As 5 0 k = k 5 k = k 4--4-6/9

()! 6. ( C0 ) + ( C ) + + ( C ) = C =!! 50 50 50 50 50 50 Ex. C 0 C + C C + + C49 C50 = C k >k +k As 49 (x+y+z)! p q x y z p! q!! p q N {0} p+q+= (x+y+z) Ex4. H 8 ( x y + z) x y z As -50 Ex5. ( x x + ) x 6 As 60 Ex6. ( x+y + z) 0 4 As 7 m! p p p ( x + x + + x ), x x x p! p! p! p, p,, p N {0} p + p + + p = m m ( x + x + + x ) m Ex7. ( x + y + z + u + v) 7 m H x y z As 0 m 9 Ex8. P( x) = + x + x + + 0x + x 0 4 8 Q ( x) = + x + 5x + + 9x + x P(x) Q(x) x 9 As 0 0 0 0 Ex9. ( x ) () x () x x As 60 80 0 Ex40. (+x)+(+x) +(+x) + +6(+x) 6 x As 89 Ex4. (+x )+(+x ) +(+x ) + +(+x ) 0 x 4 As 0 0 Ex4. = ( + ax ) x 4 660 a As ± 4--4-7/9

Ex4. ) ( kx 8 7 x x 89 k As Ex44. a R ( ax + ) 5 x 4 80 x x As 0 Ex45. ()(+x) x 5 () N (+x) x 8 x 9 x 0 () (x+) 0 495 79 As 44 4o Ex46. ( + x ) + ( + x) + ( + x) + + ( + x) x a 0 09 ()a = () = As C + a 40 = Ex47. (+x) k x P k k= 4 --- () P k () () P P As + ( ) C k = k = k k = k Ex48. N (ax+b) (bx+a) x a As Ex49. (x+)(x+)(5x+)...[(-)x+] x As ( )( 6 ) Ex50. 4 ( x ) ( x + 5 ) x 7 As 984 Ex5. + ( ) C + ( ) C + ( ) C + + ( ) C < 500 ( log=0.00 log=0.477)as 6(HINT x= y= ) Ex5. (x +x ) 5 x 6 As 90 Ex5. ) 0 0 ( + x x + ax + bx + + cx a= b= c= As a=0 b=4949 c= 4--4-8/9

Ex54. [(a-b) -c] 5 () () a b c As 6-4400 6 Ex55. (0.99) As 0.9550 4 00 Ex56. ( + 5) As 6 Ex57. 00 0 As Ex58. () x 8 (x ) As 8x 48x+ () (x+) x 00 As 4950x +9800x+4849 Ex59. 8 000 As 48 Ex60. () C + C + C + C + C + C + = As 4096 ()C ()C + C + C + 5 7 9 C C + C4 + C5 + C6 4 50 + C = As 085 + C + = As 4095 Ex6. ()C + C + C + + C 0 = As 4 () C0 + C + C + + C = As ( ) 4 ()C + C + 5C + ( + ) C = As ( + ) 0 (4) C + 5 C + 8 C + + ( ) C As ( ) + Ex6. 000< C + C + C + + C <000 = As C0 Ex6. C + C + C + + = + + = As 4 4--4-9/9