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40144299_tomtatchuongtrinhToan

40144299_tomtatchuongtrinhToan - P HAN MOT ON TAP TOM TAT...

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PHAÀN MOÄT: OÂN TAÄP TOÙM TAÉT CHÖÔNG TRÌNH THI ÑAÏI HOÏC MOÂN TOAÙN I- GIAÛI TÍCH TOÅ HÔÏP 1. Giai thöøa : n! = 1.2...n 0! = 1 n! /(n – k)! = (n – k + 1).(n – k + 2) ... n 2. Nguyeân taéc coäng : Tröôøng hôïp 1 coù m caùch choïn, tröôøng hôïp 2 coù n caùch choïn; moãi caùch choïn ñeàu thuoäc ñuùng moät tröôøng hôïp. Khi ñoù, toång soá caùch choïn laø : m + n. 3. Nguyeân taéc nhaân : Hieän töôïng 1 coù m caùch choïn, moãi caùch choïn naøy laïi coù n caùch choïn hieän töôïng 2. Khi ñoù, toång soá caùch choïn lieân tieáp hai hieän töôïng laø : m x n. 4. Hoaùn vò : Coù n vaät khaùc nhau, xeáp vaøo n choã khaùc nhau. Soá caùch xeáp : P n = n !. 5. Toå hôïp : Coù n vaät khaùc nhau, choïn ra k vaät. Soá caùch choïn : )! k n ( ! k ! n C k n - = 6. Chænh hôïp : Coù n vaät khaùc nhau. Choïn ra k vaät, xeáp vaøo k choã khaùc nhau soá caùch : = = - k k k n n n k n! A , A C .P (n k)! Chænh hôïp = toå hôïp roài hoaùn vò 7. Tam giaùc Pascal : 1 4 4 3 4 2 4 1 4 0 4 3 3 2 3 1 3 0 3 2 2 1 2 0 2 1 1 0 1 0 0 C C C C C C C C C C C C C C C 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Tính chaát : k 1 n k n 1 k n k n n k n n n 0 n C C C C C , 1 C C + - - = + = = = 8. Nhò thöùc Newton : * n 0 n n 1 1 n 1 n 0 n 0 n n b a C ... b a C b a C ) b a ( + + + = + - a = b = 1 : ... 0 1 n n n n n C C ... C 2 + + + = TRANG 1
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Vôùi a, b { ± 1, ± 2, ...}, ta chöùng minh ñöôïc nhieàu ñaúng thöùc chöùa : n n 1 n 0 n C ,..., C , C * n n n 1 n 1 n n 0 n n x C ... x a C a C ) x a ( + + + = + - Ta chöùng minh ñöôïc nhieàu ñaúng thöùc chöùa n n 1 n 0 n C ,..., C , C baèng caùch : - Ñaïo haøm 1 laàn, 2 laàn, cho x = ± 1, ± 2, ... a = ± 1, ± 2, ... - Nhaân vôùi x k , ñaïo haøm 1 laàn, 2 laàn, cho x = ± 1, ± 2, ... , a = ± 1, ± 2, ... - Cho a = ± 1, ± 2, ..., ± ± 2 0 1 0 ... hay hay β α Chuù yù : * (a + b) n : a, b chöùa x. Tìm soá haïng ñoäc laäp vôùi x : k n k k m n C a b Kx - = Giaûi pt : m = 0, ta ñöôïc k. * (a + b) n : a, b chöùa caên . Tìm soá haïng höõu tyû. m r k n k k p q n C a b Kc d - = Giaûi heä pt : Z q / r Z p / m , tìm ñöôïc k * Giaûi pt , bpt chöùa ... C , A k n k n : ñaët ñieàu kieän k, n N * ..., k n. Caàn bieát ñôn giaûn caùc giai thöøa, qui ñoàng maãu soá, ñaët thöøa soá chung. * Caàn phaân bieät : qui taéc coäng vaø qui taéc nhaân; hoaùn vò (xeáp, khoâng boác), toå hôïp (boác, khoâng xeáp), chænh hôïp (boác roài xeáp). * AÙp duïng sô ñoà nhaùnh ñeå chia tröôøng hôïp , traùnh truøng laép hoaëc thieáu tröôøng hôïp. * Vôùi baøi toaùn tìm soá caùch choïn thoûa tính chaát p maø khi chia tröôøng hôïp, ta thaáy soá caùch choïn khoâng thoûa tính chaát p ít tröôøng hôïp hôn, ta laøm nhö sau : soá caùch choïn thoûa p. = soá caùch choïn tuøy yù - soá caùch choïn khoâng thoûa p. Caàn vieát meänh ñeà phuû ñònh p thaät chính xaùc.
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