60460113 60460113 Thang Long University Libraty 11 13 16 18 1.3.7 21 24 24 24 25 26 27 28 29 30 31 2.1.9 32 33 33 Thang Long University Libraty 34 36 37 38 39 41 42 - khoa, c Thang Long University Libraty : - : - - 10 Thang Long University Libraty GI I THI U V H S NH TH C N TH C CHU N B l ch s c p trung h h p , gi i thi u v h s nh th c Nh ng ki n th c chu n v cs i thi m Minh, 2001 TP Schaum's Outline of Discrete Mathematics, McGraw - Hill, 1977 1.1 L CH S H P TRONG C P THCS: iu-t T - 1995, -t c 1.2 : n k (k k n n k k n c: n k n k n! k! n k ! n n k k k Cn n k Thang Long University Libraty T l k 2.1.6 ng nh t th c 5: n 2k 2n , ng nh t th c 6: k n n k n n , k k n a k k n b k n n k k k k n k n n ch n n k n n k 29 Thang Long University Libraty : k n k n n k n 2.1.7 ng nh t th c 7: k k n k n.2n , n a n n ? b k 0, , n k n n k k k k n k n k n k k n k n n n , n.2n 30 n 2n n k n k k n.2n n k n k 2.1.8 ng nh t th c 8: k n , n n a 2n a t p ? n Th b X: T n A n A X n k k n k 2n A {1,2, , n} : A1 1, ,k k A = {k +1, ,n- 1,n} A2 n k A1 ) A1 A2 k n k n 2 n n 31 Thang Long University Libraty n k n k n , n 2 ng nh t th c 9: k n 2.1.9 m n k k j m j n k j , m 0, n a m m n n k b m n k j 0, , k j m m j k j n n k j k j m n k k j m j n k m n j k j 32 j 2.2 NG NH T TH sau: ,N 2.2.1 ng nh t th c 10: k a n k k n 2n n i: n n n uv t h i ngh khoa h c n 33 Thang Long University Libraty b : n n 2n 2n n n 2n n k 1, , n n k k Ch k n n k n k : k V n k k ng nh t th c 10: k n 2.2.2 ng nh t th c 11: k k n k n k n k n k k n 2n n a n 2n 2n n 34 2n n b : n 2n n n : T n x n k k 0, n n k n , k 2k æ nö 2k ç ÷ è kø n k n k n k n k n k n k x x n k n : k n k k n k n k n k n k k n k n k n k n k n k 2n n 35 Thang Long University Libraty 2.2.3 n1 ng nh t th c 12: k1 k2 kr a n2 k1 k2 nr n1 n2 nr kr k : T r m1, m2 , , mr n1 m1 n2 m2 nr mr k b ? : : k1 k2 n1 m1 m2 k1 n2 k2 kr mr nr kr k1 , k2 , , kr k1 k2 kr k n1 k1 k2 kr n2 k1 k2 nr kr : k n1 n2 nr n1 n2 nr k 36 n1 k1 k2 kr 2n n2 k1 k2 nr 2n n1 n2 nr kr k 2n 2n 22 n a 2n b 2n 2n 2n 2n n 2n æ 2n ö ç 2n ÷ è 2nø 2n 2n 2n 2n 22n 2n 22 n 22 n 37 Thang Long University Libraty 2n 2n 2n 2n n 2 22 n n n n n n n.2n a n n 22 n n X x1; x2 ; ; xn a a, A X X - = X/ {a} b n a a 2n n.2n a ,A n k k 0, n n k n n k a Khi k n n k k X\A n k n n k n n n n 38 n n a ,A a n k k i n i k i 2k n k a 2k Ta n k X x1; x2 ; ; xn k b n k k 2k 2k n k n k i k k i n i n i k i n k Theo q n i k i k i k i n k n i k i 39 Thang Long University Libraty k i n k n i k i 2k n k 40 K T LU N -C - - hay 41 Thang Long University Libraty Schaum's Outline of Discrete Mathematics, McGraw - Hill, 1977 Benjamin Arthur T., and Jennifer J Quinn, Proofs that really count: the art of combinatorial proof No 27 MAA, 2003 42 - - - - trai - Thang Long University Libraty