Toan-daisotohop-chuong3 - AI SO TO HP Chng III CHNH HP Co n...

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ÑAÏI SOÁ TOÅ HÔÏP Chöông III CHÆNH HÔÏP Coù n vaät khaùc nhau, choïn ra k vaät khaùc nhau (1 k n), saép vaøo k choã khaùc nhau. Moãi caùch choïn roài saép nhö vaäy goïi laø moät chænh hôïp chaäp k cuûa n phaàn töû. Choã thöù nhaát coù n caùch choïn (do coù n vaät), choã thöù 2 coù (n – 1) caùch choïn (do coøn n – 1 vaät), choã thöù 3 coù n – 2 caùch choïn (do coøn n – 2 vaät), …, choã thöù k coù n – (k – 1) caùch choïn (do coøn n – (k – 1) vaät). Vaäy, theo qui taéc nhaân, soá caùch choïn laø : n × (n – 1) × (n – 2) × × (n – k + 1) = n! (n k)! Neáu kí hieäu soá chænh hôïp chaäp k cuûa n phaàn töû laø , ta coù : k n A = k n A Ví duï 1. Moät nhaø haøng coù 5 moùn aên chuû löïc, caàn choïn 2 moùn aên chuû löïc khaùc nhau cho moãi ngaøy, moät moùn buoåi tröa vaø moät moùn buoåi chieàu. Hoûi coù maáy caùch choïn ? Giaûi Ñaây laø chænh hôïp chaäp 2 cuûa 5 phaàn töû, coù : = 2 5 A 5! (5 2)! = 4.5 = 20 caùch choïn. (Giaû söû 5 moùn aên ñöôïc ñaùnh soá 1, 2, 3, 4, 5; ta coù caùc caùch choïn sau ñaây : (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4)). Ví duï 2. Trong moät tröôøng ñaïi hoïc, ngoaøi caùc moân hoïc baét buoäc, coù 3 moân töï choïn, sinh vieân phaûi choïn ra 2 moân trong 3 moân ñoù, 1 moân chính vaø 1 moân phuï. Hoûi coù maáy caùch choïn ? Giaûi Ñaây laø chænh hôïp chaäp 2 cuûa 3 phaàn töû. Vaäy coù :
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= 2 3 A 3! (3 2)! = 6 caùch choïn. (Giaû söû 3 moân töï choïn laø a, b, c thì 6 caùch choïn theo yeâu caàu laø (a, b), (a, c), (b, a), (b, c), (c, a), (c, b)). Ví duï 3. Töø 5 chöõ soá 1, 2, 3, 4, 5 coù theå taïo ra bao nhieâu soá goàm 2 chöõ soá khaùc nhau ? Giaûi Ñaây laø chænh hôïp chaäp 2 cuûa 5 phaàn töû. Vaäy coù : = 2 5 A 5! (5 2)! = 5! 3! = 5 × 4 = 20 soá (Caùc soá ñoù laø : 12, 13, 14, 15, 21, 23, 24, 25, 31, 32, 34, 35, 41, 42, 43, 45, 51, 52, 53, 54) . Baøi 35. Chöùng minh vôùi n, k vaø 2 ¥ k < n a) k n A = k n1 A + k k1 A b) n2 nk A + + + A + + = k 2 n A + Giaûi a) Ta coù : k A + k A = (n 1)! (n 1 k)! −− + k. (n 1)! (n = (n – 1)! 1k (n k 1)! (n k)(n k 1)! + = (n 1)! (n k 1)! k 1 ⎛⎞ + ⎜⎟ ⎝⎠ = (n 1)! (n k 1)! . n = n! (n = k n A . b) A + + + A + + = (n (k 2)! + + (n (k 1)! + = (n (k + + (n (k 1)(k + = (n (k + 1 1 + = (n (k + . k = 2 (n k)!k k! + = n A + .k 2 . Baøi 36. Giaûi phöông trình P x . 2 x A + 72 = 6( 2 x A + 2P x ).
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Ñaïi hoïc Quoác gia Haø Noäi khoái D 2001 Giaûi Ñieàu kieän x vaø x 2. ¥ Ta coù : P x . 2 x A + 72 = 6( 2 x A + 2P x ) x! x! (x 2)! + 72 = 6 2x! + x!x(x – 1) + 72 = 6[x(x – 1) + 2x!] (x 2 – x – 12)x! = 6(x 2 – x – 12) (x 2 – x – 12)(x! – 6) = 0 2 xx 1 2 x! 6 0 −− = −= 0 3 : loaïi x4 x x3 = =− = = = Baøi 37. Giaûi baát phöông trình : 3 A x + 5 2 x A 21x.
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Toan-daisotohop-chuong3 - AI SO TO HP Chng III CHNH HP Co n...

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