{"id":306928,"date":"2024-01-14T12:25:20","date_gmt":"2024-01-14T05:25:20","guid":{"rendered":"https:\/\/hoanghamobile.com\/tin-tuc\/?p=306928"},"modified":"2024-01-14T12:25:20","modified_gmt":"2024-01-14T05:25:20","slug":"cong-thuc-chinh-hop","status":"publish","type":"post","link":"https:\/\/hoanghamobile.com\/tin-tuc\/cong-thuc-chinh-hop\/","title":{"rendered":"C\u00f4ng th\u1ee9c t\u00ednh ho\u00e1n v\u1ecb, ch\u1ec9nh h\u1ee3p v\u00e0 t\u1ed5 h\u1ee3p \u0111\u1ea7y \u0111\u1ee7"},"content":{"rendered":"

Kh\u00f4ng \u00edt ng\u01b0\u1eddi khi ti\u1ebfp x\u00fac v\u1edbi <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b>, ho\u00e1n v\u1ecb v\u00e0 t\u1ed5 h\u1ee3p s\u1ebd b\u1ecb nh\u1ea7m l\u1eabn kh\u00e1i ni\u1ec7m, c\u0169ng nh\u01b0 ph\u00e2n bi\u1ec7t c\u00f4ng th\u1ee9c ch\u00ednh x\u00e1c. Ch\u00ednh v\u00ec th\u1ebf, Ho\u00e0ng H\u00e0 Mobile \u0111\u00e3 t\u1ed5ng h\u1ee3p b\u00e0i vi\u1ebft n\u00e0y \u0111\u1ec3 gi\u1ea3i th\u00edch r\u00f5 h\u01a1n v\u1ec1 b\u1ed9 3 c\u00f4ng th\u1ee9c to\u00e1n h\u1ecdc tr\u00ean. B\u00ean c\u1ea1nh \u0111\u00f3, ch\u00fang t\u00f4i c\u0169ng c\u00f3 v\u00ed d\u1ee5 minh h\u1ecda \u1edf t\u1eebng c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u00fap b\u1ea1n hi\u1ec3u r\u00f5 h\u01a1n.<\/span><\/p>\n

Ho\u00e1n v\u1ecb l\u00e0 g\u00ec?<\/b><\/h2>\n

Ho\u00e1n v\u1ecb l\u00e0 m\u1ed9t kh\u00e1i ni\u1ec7m quan tr\u1ecdng trong to\u00e1n h\u1ecdc v\u00e0 l\u00fd thuy\u1ebft \u0111\u1ed3 th\u1ecb. N\u00f3 \u0111\u01b0\u1ee3c d\u00f9ng \u0111\u1ec3 m\u00f4 t\u1ea3 c\u00e1ch m\u00e0 c\u00e1c ph\u1ea7n t\u1eed c\u1ee7a m\u1ed9t t\u1eadp h\u1ee3p c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp l\u1ea1i ho\u1eb7c \u0111\u1ed5i ch\u1ed7 v\u1edbi nhau \u0111\u1ec3 t\u1ea1o ra m\u1ed9t s\u1ef1 thay \u0111\u1ed5i trong c\u1ea5u tr\u00fac ban \u0111\u1ea7u. \u0110\u00e2y l\u00e0 m\u1ed9t kh\u00eda c\u1ea1nh c\u1ee7a l\u00fd thuy\u1ebft t\u1ed5 h\u1ee3p, n\u01a1i nghi\u00ean c\u1ee9u v\u1ec1 s\u1ef1 s\u1eafp x\u1ebfp, k\u1ebft h\u1ee3p v\u00e0 ch\u1ecdn l\u1ef1a c\u1ee7a c\u00e1c ph\u1ea7n t\u1eed trong m\u1ed9t t\u1eadp h\u1ee3p h\u1eefu h\u1ea1n.<\/span><\/p>\n

\"cong-thuc-chinh-hop-1\"<\/picture><\/p>\n

M\u1ed9t ho\u00e1n v\u1ecb c\u1ee7a m\u1ed9t t\u1eadp h\u1ee3p g\u1ed3m c\u00e1c ph\u1ea7n t\u1eed \u0111\u01b0\u1ee3c x\u1ebfp theo m\u1ed9t th\u1ee9 t\u1ef1 c\u1ee5 th\u1ec3. S\u1ef1 kh\u00e1c bi\u1ec7t gi\u1eefa ho\u00e1n v\u1ecb v\u00e0 c\u00e1c kh\u00e1i ni\u1ec7m kh\u00e1c nh\u01b0 t\u1ed5 h\u1ee3p v\u00e0 ch\u00edn ph\u1ea7n t\u1eed c\u00f2n l\u1ea1i c\u1ee7a t\u1eadp h\u1ee3p, l\u00e0 m\u1ed7i ph\u1ea7n t\u1eed ch\u1ec9 xu\u1ea5t hi\u1ec7n m\u1ed9t l\u1ea7n duy nh\u1ea5t trong m\u1ed7i ho\u00e1n v\u1ecb. \u0110i\u1ec1u n\u00e0y t\u1ea1o ra m\u1ed9t s\u1ed1 l\u01b0\u1ee3ng ho\u00e1n v\u1ecb kh\u00e1c nhau t\u00f9y thu\u1ed9c v\u00e0o s\u1ed1 l\u01b0\u1ee3ng ph\u1ea7n t\u1eed trong t\u1eadp h\u1ee3p.<\/span><\/p>\n

Ch\u1eb3ng h\u1ea1n nh\u01b0 ta c\u00f3 m\u1ed9t t\u1eadp h\u1ee3p A g\u1ed3m c\u00f3 N ph\u1ea7n t\u1eed v\u1edbi N l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 0. M\u1ed7i c\u00e1ch s\u1eafp x\u1ebfp c\u1ee7a N ph\u1ea7n t\u1eed theo m\u1ed9t v\u1ecb tr\u00ed c\u1ee5 th\u1ec3 n\u00e0o \u0111\u00f3 s\u1ebd \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 ho\u00e1n v\u1ecb c\u1ee7a N ph\u1ea7n t\u1eed. T\u01b0\u01a1ng \u1ee9ng s\u1ed1 th\u1ee9 t\u1ef1 ho\u00e1n v\u1ecb c\u1ee7a N s\u1ebd \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u l\u00e0 Pn.<\/span><\/p>\n

T\u1ed5 h\u1ee3p l\u00e0 g\u00ec?<\/b><\/h2>\n

Trong b\u1ed9 3 <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b>, t\u1ed5 h\u1ee3p v\u00e0 ho\u00e1n v\u1ecb, th\u00ec t\u1ed5 h\u1ee3p v\u00e0 ho\u00e1n v\u1ecb c\u00f3 kh\u00e1i ni\u1ec7m kh\u00e1 t\u01b0\u01a1ng \u0111\u1ed3ng nhau. Ch\u00ednh v\u00ec v\u1eady \u0111\u00e3 l\u00e0m cho kh\u00f4ng \u00edt ng\u01b0\u1eddi hi\u1ec3u nh\u1ea7m l\u1eabn, c\u0169ng nh\u01b0 v\u1eadn d\u1ee5ng sai c\u00e1ch trong vi\u1ec7c t\u00ednh to\u00e1n. Hi\u1ec3u \u0111\u01b0\u1ee3c \u0111i\u1ec1u n\u00e0y n\u00ean Ho\u00e0ng H\u00e0 Mobile \u0111\u00e3 th\u1ed1ng k\u00ea \u0111\u01b0\u1ee3c kh\u00e1i ni\u1ec7m ch\u00ednh x\u00e1c c\u1ee7a t\u1ed5 h\u1ee3p.<\/span><\/p>\n

Trong \u0111\u00f3, t\u1ed5 h\u1ee3p l\u00e0 c\u00e1ch c\u00e1c ph\u1ea7n t\u1eed c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp, k\u1ebft h\u1ee3p v\u00e0 ch\u1ecdn l\u1ef1a t\u1eeb m\u1ed9t t\u1eadp h\u1ee3p h\u1eefu h\u1ea1n \u0111\u1ec3 t\u1ea1o ra c\u00e1c c\u1ea5u tr\u00fac m\u1edbi. Hay n\u00f3i theo c\u00e1ch kh\u00e1c, \u0111\u00e2y l\u00e0 c\u00e1ch ch\u00fang ta l\u1ef1a ch\u1ecdn ra c\u00e1c ph\u1ea7n t\u1eed t\u1eeb m\u1ed9t t\u1eadp h\u1ee3p X l\u1edbn m\u00e0 kh\u00f4ng ph\u00e2n bi\u1ec7t v\u1ecb tr\u00ed c\u1ee7a ch\u00fang n\u1eb1m \u1edf \u0111\u00e2u. \u0110\u1eb7c bi\u1ec7t v\u1edbi m\u1ed9t s\u1ed1 tr\u01b0\u1eddng h\u1ee3p ngo\u1ea1i l\u1ec7, b\u1ea1n c\u00f2n c\u00f3 th\u1ec3 \u0111\u1ebfm \u0111\u01b0\u1ee3c s\u1ed1 t\u1ed5 h\u1ee3p.<\/span><\/p>\n

\"cong-thuc-chinh-hop-2\"<\/picture><\/p>\n

H\u01a1n n\u1eefa, trong t\u1ed5 h\u1ee3p c\u00e1c kh\u00e1i ni\u1ec7m nh\u01b0 t\u1ed5 h\u1ee3p ch\u1eadp k v\u00e0 t\u1ed5 h\u1ee3p nh\u1ecb th\u1ee9c \u0111\u1ec1u \u0111\u00f3ng vai tr\u00f2 quan tr\u1ecdng. T\u1ed5 h\u1ee3p ch\u1eadp k m\u00f4 t\u1ea3 c\u00e1ch l\u1ef1a ch\u1ecdn k ph\u1ea7n t\u1eed t\u1eeb m\u1ed9t t\u1eadp h\u1ee3p. Trong khi t\u1ed5 h\u1ee3p nh\u1ecb th\u1ee9c th\u01b0\u1eddng \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 \u0111\u1ebfm s\u1ed1 c\u00e1ch kh\u00e1c nhau m\u00e0 m\u1ed9t s\u1ef1 ki\u1ec7n c\u00f3 th\u1ec3 x\u1ea3y ra.<\/span><\/p>\n

V\u1eadn d\u1ee5ng v\u00e0o to\u00e1n h\u1ecdc, ch\u00fang ta s\u1ebd d\u1ec5 h\u00ecnh dung h\u01a1n khi n\u00f3i v\u1edbi m\u1ed7i t\u1eadp h\u1ee3p con bao h\u00e0m k ph\u1ea7n t\u1eed v\u1edbi a > 0 s\u1ebd \u0111\u01b0\u1ee3c xem l\u00e0 m\u1ed9t t\u1ed5 h\u1ee3p ch\u1eadp k c\u1ee7a a.<\/span><\/p>\n

Ch\u1ec9nh h\u1ee3p l\u00e0 g\u00ec?<\/b><\/h2>\n

Ti\u1ebfp theo \u0111\u00e2y, ch\u00fang ta s\u1ebd c\u00f9ng nhau t\u00ecm hi\u1ec3u chi ti\u1ebft h\u01a1n v\u1ec1 kh\u00e1i ni\u1ec7m <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b> trong to\u00e1n h\u1ecdc. Ch\u1ec9nh h\u1ee3p m\u00f4 t\u1ea3 c\u00e1ch c\u00e1c ph\u1ea7n t\u1eed trong m\u1ed9t t\u1eadp h\u1ee3p c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ch\u1ecdn v\u00e0 s\u1eafp x\u1ebfp theo m\u1ed9t th\u1ee9 t\u1ef1 c\u1ee5 th\u1ec3 \u0111\u1ec3 t\u1ea1o ra c\u00e1c c\u1ea5u tr\u00fac kh\u00e1c nhau. Kh\u00e1c v\u1edbi ho\u00e1n v\u1ecb, ch\u1ec9nh h\u1ee3p kh\u00f4ng y\u00eau c\u1ea7u r\u1eb1ng t\u1ea5t c\u1ea3 c\u00e1c ph\u1ea7n t\u1eed \u0111\u1ec1u ph\u1ea3i \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng, m\u00e0 ch\u1ec9 l\u00e0 m\u1ed9t ph\u1ea7n c\u1ee7a ch\u00fang. Ch\u1ec9nh h\u1ee3p th\u01b0\u1eddng \u0111\u01b0\u1ee3c th\u01b0\u1eddng \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u l\u00e0 A(n,k), v\u1edbi n l\u00e0 s\u1ed1 ph\u1ea7n t\u1eed trong t\u1ed5 h\u1ee3p v\u00e0 k l\u00e0 s\u1ed1 ph\u1ea7n t\u1eed \u0111\u01b0\u1ee3c ch\u1ecdn.<\/span><\/p>\n

\"cong-thuc-chinh-hop-3\"<\/picture><\/p>\n

C\u00f4ng th\u1ee9c to\u00e1n h\u1ecdc ch\u1ec9nh h\u1ee3p c\u00f3 nhi\u1ec1u \u1ee9ng d\u1ee5ng trong th\u1ef1c t\u1ebf. V\u00ed d\u1ee5, trong m\u1eadt m\u00e3 h\u1ecdc, n\u00f3 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh s\u1ed1 l\u01b0\u1ee3ng kh\u00f3a m\u00e3 h\u00f3a kh\u00e1c nhau. Trong qu\u1ea3n l\u00fd d\u1ef1 \u00e1n, ch\u1ec9nh h\u1ee3p c\u00f3 th\u1ec3 \u00e1p d\u1ee5ng \u0111\u1ec3 x\u00e1c \u0111\u1ecbnh s\u1ed1 c\u00e1ch kh\u00e1c nhau \u0111\u1ec3 l\u1eadp l\u1ecbch c\u00f4ng vi\u1ec7c ho\u1eb7c x\u1ebfp h\u00e0ng t\u00e1c v\u1ee5. Ngo\u00e0i ra, trong l\u0129nh v\u1ef1c x\u00e1c su\u1ea5t, ch\u1ec9nh h\u1ee3p th\u01b0\u1eddng xu\u1ea5t hi\u1ec7n khi xem x\u00e9t c\u00e1c s\u1ef1 ki\u1ec7n c\u00f3 th\u1ee9 t\u1ef1 c\u1ee5 th\u1ec3.<\/span><\/p>\n

C\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p, ho\u00e1n v\u1ecb v\u00e0 t\u1ed5 h\u1ee3p c\u00f3 m\u1ed1i quan h\u1ec7 nh\u01b0 th\u1ebf n\u00e0o?<\/b><\/h2>\n

V\u1edbi ph\u1ea7n kh\u00e1i ni\u1ec7m m\u00e0 Ho\u00e0ng H\u00e0 Mobile v\u1eeba chia s\u1ebb, ch\u00fang ta th\u1ea5y \u0111\u01b0\u1ee3c m\u1ed1i li\u00ean k\u1ebft s\u00e2u s\u1eafc gi\u1eefa t\u1ed5 h\u1ee3p, ho\u00e1n v\u1ecb v\u00e0 ch\u1ec9nh h\u1ee3p. Theo \u0111\u00f3, m\u1ed9t ch\u1ec9nh h\u1ee3p ch\u1eadp k c\u1ee7a n s\u1ebd \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh d\u1ef1a tr\u00ean 2 thao t\u00e1c x\u00e1c \u0111\u1ecbnh c\u01a1 b\u1ea3n.\u00a0<\/span><\/p>\n

(1) \u0110\u1ea7u ti\u00ean, b\u1ea1n c\u1ea7n l\u1ea5y m\u1ed9t t\u1ed5 h\u1ee3p ch\u1eadp k c\u1ee7a n ph\u1ea7n t\u1eed.<\/span><\/p>\n

(2) Sau \u0111\u00f3, b\u1ea1n s\u1ebd th\u1ef1c hi\u1ec7n ho\u00e1n v\u1ecb k ph\u1ea7n t\u1eed l\u00e0 \u0111\u00e3 t\u1ea1o th\u00e0nh m\u1ed9t ch\u1ec9nh h\u1ee3p ch\u1eadp k c\u1ee7a n.<\/span><\/p>\n

\"cong-thuc-chinh-hop-4\"<\/picture><\/p>\n

C\u00f3 th\u1ec3 n\u00f3i, nh\u1eefng <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b>, ho\u00e1n v\u1ecb v\u00e0 t\u1ed5 h\u1ee3p s\u1ebd xu\u1ea5t hi\u1ec7n th\u01b0\u1eddng xuy\u00ean trong nh\u1eefng b\u00e0i to\u00e1n THPT. \u0110\u1eb7c bi\u1ec7t n\u00f3 c\u00f2n n\u1eb1m trong \u0111\u1ec1 thi t\u1ed1t nghi\u1ec7p THPT nh\u1eefng n\u0103m v\u1eeba qua. Ch\u00ednh v\u00ec l\u1ebd \u0111\u00f3, b\u1ea1n c\u1ea7n hi\u1ec3u r\u00f5 h\u01a1n v\u1ec1 b\u1ea3n ch\u1ea5t, c\u0169ng nh\u01b0 c\u00f4ng th\u1ee9c \u0111\u1ec3 \u00e1p d\u1ee5ng theo \u0111\u00fang t\u1eebng tr\u01b0\u1eddng h\u1ee3p kh\u00e1c nhau.<\/span><\/p>\n

H\u01b0\u1edbng d\u1eabn quy t\u1eafc \u0111\u1ebfm ho\u00e1n v\u1ecb, t\u1ed5 h\u1ee3p v\u00e0 ch\u1ec9nh h\u1ee3p \u0111\u01a1n gi\u1ea3n nh\u1ea5t<\/b><\/h2>\n

Ti\u1ebfp t\u1ee5c n\u1ed9i dung, Ho\u00e0ng H\u00e0 Mobile s\u1ebd chia s\u1ebb th\u00eam \u0111\u1ebfn b\u1ea1n m\u1ed9t b\u1ed9 quy t\u1eafc \u0111\u1ebfm c\u1ef1c k\u1ef3 quan tr\u1ecdng \u0111\u1ec3 s\u1eed d\u1ee5ng cho c\u00f4ng th\u1ee9c ho\u00e1n v\u1ecb, t\u1ed5 h\u1ee3p v\u00e0 ch\u1ec9nh h\u1ee3p. Trong \u0111\u00f3, m\u1ed7i kh\u00e1i ni\u1ec7m s\u1ebd c\u00f3 m\u1ed9t quy t\u1eafc \u0111\u1ebfm kh\u00e1c nhau. Do v\u1eady, b\u1ea1n \u0111\u1eebng b\u1ecf qua ph\u1ea7n n\u1ed9i dung quan tr\u1ecdng n\u00e0y b\u1ea1n nh\u00e9.<\/span><\/p>\n

Quy t\u1eafc \u0111\u1ebfm ho\u00e1n v\u1ecb<\/b><\/h3>\n

Nh\u01b0 ch\u00fang ta \u0111\u01b0\u1ee3c bi\u1ebft, ho\u00e1n v\u1ecb m\u00f4 t\u1ea3 v\u1ec1 m\u1ed9t t\u1eadp h\u1ee3p k ph\u1ea7n t\u1eed. V\u00ec th\u1ebf, t\u1eeb t\u1eadp h\u1ee3p k ph\u1ea7n t\u1eed n\u00e0y ch\u00fang ta c\u00f3 th\u1ec3 thi\u1ebft l\u1eadp \u0111\u01b0\u1ee3c ho\u00e1n v\u1ecb c\u1ee7a a ph\u1ea7n t\u1eed v\u1edbi hai thao t\u00e1c si\u00eau \u0111\u01a1n gi\u1ea3n.\u00a0<\/span><\/p>\n

(1) \u0110\u1ea7u ti\u00ean, b\u1ea1n s\u1ebd ch\u1ecdn ph\u1ea7n t\u1eed \u0111\u1ea7u ti\u00ean v\u00e0 th\u1ef1c hi\u1ec7n c\u1ed9ng t\u1ed5ng n c\u00e1ch.<\/span><\/p>\n

(2) Sau \u0111\u00f3, b\u1ea1n s\u1ebd ch\u1ecdn ph\u1ea7n t\u1eed ti\u1ebfp theo v\u00e0 th\u1ef1c hi\u1ec7n c\u00f4ng th\u1ee9c <\/span>(n-1)<\/b> theo c\u00e1ch s\u1eafp x\u1ebfp ho\u00e1n v\u1ecb.<\/span><\/p>\n

(3) C\u1ee9 ti\u1ebfp t\u1ee5c nh\u01b0 v\u1eady, ch\u00fang ta s\u1ebd th\u1ef1c hi\u1ec7n \u0111\u1ebfn c\u00e1c ph\u1ea7n t\u1eed ti\u1ebfp theo v\u00e0 \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c chung l\u00e0 <\/span>(a-1)<\/b>.<\/span><\/p>\n

\"cong-thuc-chinh-hop-5\"<\/picture><\/p>\n

L\u01b0u \u00fd: C\u00f3 m\u1ed9t s\u1ed1 tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t khi n = a, ch\u00fang ta s\u1ebd \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c kh\u00e1c \u0111\u1ec3 t\u00ednh s\u1ed1 c\u00e1c ho\u00e1n v\u1ecb kh\u00f4ng gi\u1ed1ng nhau l\u00e0 P(n) = n!. C\u00f2n n\u1ebfu trong tr\u01b0\u1eddng h\u1ee3p a < n th\u00ec s\u1ed1 ho\u00e1n v\u1ecb s\u1ebd \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c l\u00e0 <\/span>P(n,a) = n!(n-a)!<\/b>.<\/span><\/p>\n

Quy t\u1eafc \u0111\u1ebfm t\u1ed5 h\u1ee3p<\/b><\/h3>\n

Tr\u01b0\u1edbc khi b\u1ea1n mu\u1ed1n bi\u1ebft c\u00e1ch t\u00ednh <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b>, ho\u00e1n v\u1ecb v\u00e0 t\u1ed5 h\u1ee3p th\u00ec b\u1ea1n ph\u1ea3i hi\u1ec3u \u0111\u01b0\u1ee3c quy t\u1eafc \u0111\u1ebfm c\u1ee7a n\u00f3. V\u00e0 ti\u1ebfp t\u1ee5c n\u1ed9i dung b\u00e0i vi\u1ebft, ch\u00fang ta s\u1ebd c\u00f9ng t\u00ecm hi\u1ec3u v\u1ec1 quy t\u1eafc \u0111\u1ebfm t\u1ed5 h\u1ee3p. C\u00f3 th\u1ec3 n\u00f3i, t\u1ed5 h\u1ee3p c\u00f3 ph\u01b0\u01a1ng th\u1ee9c \u0111\u1ebfm \u0111\u01a1n gi\u1ea3n nh\u1ea5t v\u00e0 ai c\u0169ng c\u00f3 th\u1ec3 th\u1ef1c hi\u1ec7n khi \u0111\u1ecdc qua chia s\u1ebb c\u1ee7a ch\u00fang t\u00f4i.<\/span><\/p>\n

\"cong-thuc-chinh-hop-6\"<\/picture><\/p>\n

Ch\u00fang ta s\u1ebd c\u00f3 m\u1ed9t t\u1ed5 h\u1ee3p X bao g\u1ed3m n ph\u1ea7n t\u1eed v\u1edbi n l\u1edbn h\u01a1n 0. Ti\u1ebfp \u0111\u1ebfn, ch\u00fang ta s\u1ebd c\u00f3 m\u1ed9t t\u1ed5 h\u1ee3p ch\u1eadp k v\u1edbi c\u00e1c ph\u1ea7n t\u1eed b\u1ea5t k\u1ef3 thu\u1ed9c t\u1ed5 h\u1ee3p X. Trong \u0111\u00f3, t\u1ed5 h\u1ee3p con \u0111\u00e1p \u1ee9ng \u0111\u01b0\u1ee3c c\u00e1c \u0111i\u1ec1u ki\u1ec7n v\u1ec1 X; 0 \u2264 k \u2264 n ; k \u2208 n. T\u1eeb \u0111\u00e2y, ch\u00fang ta s\u1ebd c\u00f3 \u0111\u01b0\u1ee3c c\u00f4ng th\u1ee9c c\u1ee7a quy t\u1eafc \u0111\u1ebfm t\u1ed5 h\u1ee3p b\u1eb1ng n!(n-k)!<\/strong>.\u00a0<\/span><\/p>\n

Quy t\u1eafc \u0111\u1ebfm ch\u1ec9nh h\u1ee3p<\/b><\/h3>\n

C\u00f3 th\u1ec3 n\u00f3i, c\u00f4ng th\u1ee9c \u0111\u1ebfm ho\u00e1n v\u1ecb l\u00e0 c\u00f4ng th\u1ee9c chung cho c\u1ea3 t\u1ed5 h\u1ee3p v\u00e0 ch\u1ec9nh h\u1ee3p. Ch\u00ednh v\u00ec v\u1eady, n\u1ebfu b\u1ea1n \u0111\u00e3 hi\u1ec3u r\u00f5 \u0111\u01b0\u1ee3c ho\u00e1n v\u1ecb th\u00ec c\u00e1ch t\u00ednh c\u1ee7a t\u1ed5 h\u1ee3p v\u00e0 ch\u1ec9nh h\u1ee3p c\u0169ng r\u1ea5t \u0111\u01a1n gi\u1ea3n. C\u1ee5 th\u1ec3 nh\u01b0 trong ph\u1ea7n n\u00e0y l\u00e0 th\u1ee7 thu\u1eadt \u0111\u1ebfm c\u1ee7a ch\u1ec9nh h\u1ee3p.<\/span><\/p>\n

\"cong-thuc-chinh-hop-7\"<\/picture><\/p>\n

Quy t\u1eafc \u0111\u1ebfm cho ch\u1ec9nh h\u1ee3p \u0111\u01b0\u1ee3c m\u00f4 t\u1ea3 b\u1eb1ng m\u1ed9t t\u1eadp h\u1ee3p l\u1edbn X v\u1edbi n\u2265 1, n l\u00e0 s\u1ed1 ph\u1ea7n t\u1eed trong t\u1eadp h\u1ee3p, k l\u00e0 s\u1ed1 l\u01b0\u1ee3ng ph\u1ea7n t\u1eed \u0111\u01b0\u1ee3c ch\u1ecdn v\u00e0 s\u1eafp x\u1ebfp. \u0110i k\u00e8m v\u1edbi \u0111\u00f3 l\u00e0 m\u1ed9t d\u00e3y \u0111i\u1ec1u ki\u1ec7n v\u1ec1 1 \u2264 k \u2264 n ; k \u2208 n. T\u1eeb \u0111\u00e2y, ch\u00fang ta s\u1ebd c\u00f3 \u0111\u01b0\u1ee3c c\u00f4ng th\u1ee9c \u0111\u1ebfm c\u1ee7a ch\u1ec9nh h\u1ee3p b\u1eb1ng n!k!(n-k)!<\/strong>.<\/span><\/p>\n

C\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p, t\u1ed5 h\u1ee3p, ho\u00e1n v\u1ecb v\u00e0 v\u00ed d\u1ee5 minh h\u1ecda<\/b><\/h2>\n

Sau khi ch\u00fang ta \u0111\u00e3 t\u00ecm hi\u1ec3u \u0111\u01b0\u1ee3c c\u00e1c ph\u1ea7n \u201cr\u00eca\u201d c\u1ee7a l\u00fd thuy\u1ebft, ngay sau \u0111\u00e2y l\u00e0 ph\u1ea7n n\u1ed9i dung quan tr\u1ecdng nh\u1ea5t b\u00e0i vi\u1ebft n\u00e0y. B\u1ea1n \u0111\u1ecdc h\u00e3y c\u00f9ng Ho\u00e0ng H\u00e0 Mobile h\u1ecdc thu\u1ed9c c\u00f4ng th\u1ee9c t\u00ednh ch\u1ec9nh h\u1ee3p, t\u1ed5 h\u1ee3p v\u00e0 ho\u00e1n v\u1ecb nh\u00e9. T\u1ea5t nhi\u00ean \u201ch\u1ecdc s\u1ebd \u0111i \u0111\u00f4i v\u1edbi h\u00e0nh\u201d v\u00e0 ch\u00fang t\u00f4i c\u0169ng chia s\u1ebb cho b\u1ea1n m\u1ed9t v\u00e0i v\u00ed d\u1ee5 minh h\u1ecda c\u1ee5 th\u1ec3.<\/span><\/p>\n

C\u00f4ng th\u1ee9c ho\u00e1n v\u1ecb<\/b><\/h3>\n

C\u00f4ng th\u1ee9c t\u00ednh ho\u00e1n v\u1ecb c\u1ee7a n ph\u1ea7n t\u1eed ph\u1ea3i n\u00f3i l\u00e0 c\u1ef1c k\u1ef3 gi\u1ea3n \u0111\u01a1n. Ch\u00fang ta \u0111\u00e3 bi\u1ebft \u0111\u01b0\u1ee3c k\u00fd hi\u1ec7u c\u1ee7a ho\u00e1n v\u1ecb l\u00e0 P(n). V\u1eady ch\u00fang ta s\u1ebd c\u00f3 \u0111\u01b0\u1ee3c c\u00f4ng th\u1ee9c t\u00ednh ho\u00e1n v\u1ecb l\u00e0 <\/span>P(n) = n!<\/b>. Trong \u0111\u00f3, n! bi\u1ec3u th\u1ecb giai th\u1eeba c\u1ee7a nk, t\u1ee9c l\u00e0 t\u00edch c\u1ee7a t\u1ea5t c\u1ea3 c\u00e1c s\u1ed1 nguy\u00ean t\u1eeb 1 \u0111\u1ebfn n.<\/span><\/p>\n

\"cong-thuc-chinh-hop-8\"<\/picture><\/p>\n

V\u00ed d\u1ee5: B\u00e0i to\u00e1n c\u1ee7a ch\u00fang l\u00e0 c\u00f3 m\u1ed9t t\u1eadp h\u1ee3p g\u1ed3m 4 ph\u1ea7n t\u1eed X = {A, B, C, D} = {4, 3, 2, 1}. V\u1eady t\u1eeb t\u1eadp h\u1ee3p X ch\u00fang ta c\u00f3 th\u1ec3 l\u1eadp ra bao nhi\u00eau s\u1ed1 g\u1ed3m c\u00f3 4 ch\u1eef s\u1ed1 ph\u00e2n bi\u1ec7t?<\/span><\/p>\n

\u0110\u00e1p \u00e1n: B\u1ea1n \u0111\u1ecdc h\u00e3y \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c P(n) = n! = P(4) = 4! = 4\u00d73\u00d72\u00d71= 24. Suy ra ch\u00fang ta s\u1ebd c\u00f3 24 c\u00e1ch s\u1eafp x\u1ebfp ph\u1ea7n t\u1eed kh\u00e1c nhau t\u1eeb t\u1eadp h\u1ee3p X.<\/span><\/p>\n

C\u00f4ng th\u1ee9c t\u1ed5 h\u1ee3p<\/b><\/h3>\n

\u0110\u1ed1i v\u1edbi c\u00f4ng th\u1ee9c t\u00ednh t\u1ed5 h\u1ee3p s\u1ebd t\u01b0\u01a1ng \u0111\u1ed1i kh\u00f3 h\u01a1n <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p <\/b>v\u00e0 ho\u00e1n v\u1ecb. C\u1ee5 th\u1ec3 c\u00f4ng th\u1ee9c t\u00ednh t\u1ed5 h\u1ee3p c\u1ee7a n ph\u1ea7n t\u1eed ch\u1ecdn k ph\u1ea7n t\u1eed, k\u00fd hi\u1ec7u C(n,k) l\u00e0: <\/span>C(n,k) = n! \/ (k!(n-k)! = (n(n-1)(n-2)…(n-k+1))\/k!. <\/b>Trong c\u00f4ng th\u1ee9c n\u00e0y, n! bi\u1ec3u th\u1ecb giai th\u1eeba c\u1ee7a n, k! l\u00e0 giai th\u1eeba c\u1ee7a k v\u00e0 (n-k)! l\u00e0 giai th\u1eeba c\u1ee7a n-k.<\/span><\/p>\n

\"cong-thuc-chinh-hop-9\"<\/picture><\/p>\n

V\u00ed d\u1ee5, n\u1ebfu b\u1ea1n c\u00f3 m\u1ed9t t\u1eadp h\u1ee3p g\u1ed3m 5 ph\u1ea7n t\u1eed {A, B, C, D, E} = {5, 4, 3, 2, 1} v\u00e0 b\u1ea1n mu\u1ed1n ch\u1ecdn 3 ph\u1ea7n t\u1eed, s\u1ed1 t\u1ed5 h\u1ee3p c\u1ee7a ch\u00fang l\u00e0:C(5,3) = 5! \/ 3!(5\u22123) = (5\u00d74\u00d73) \/ (3\u00d72\u00d71) = 10.<\/span><\/p>\n

C\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b><\/h3>\n

C\u00f4ng th\u1ee9c t\u00ednh ch\u1ec9nh h\u1ee3p c\u0169ng kh\u00e1 \u0111\u01a1n gi\u1ea3n, b\u1ea1n ch\u1ec9 c\u1ea7n \u0111\u1ecdc hi\u1ec3u nh\u1eefng d\u1eef ki\u1ec7n trong b\u00e0i to\u00e1n v\u1edbi n ph\u1ea7n t\u1eed ch\u1ecdn k ph\u1ea7n t\u1eed v\u00e0 s\u1eafp x\u1ebfp theo m\u1ed9t th\u1ee9 t\u1ef1 c\u1ee5 th\u1ec3. \u0110\u1ed3ng th\u1eddi, b\u1ea1n s\u1ebd \u00e1p d\u1ee5ng c\u00f4ng th\u1ee9c t\u00ednh ch\u1ec9nh h\u1ee3p m\u00e0 ch\u00fang t\u00f4i cung c\u1ea5p b\u1eb1ng <\/span>A (n,k) = n! \/ (n-k)!<\/b> =\u00a0 n(n-1)(n-2)…(n-k+1)<\/b>.<\/span><\/p>\n

\"cong-thuc-chinh-hop-10\"<\/picture><\/p>\n

V\u00ed d\u1ee5: B\u00e0i to\u00e1n cho m\u1ed9t t\u1eadp h\u1ee3p g\u1ed3m 6 ph\u1ea7n t\u1eed {A, B, C, D, E, F} = {6, 5, 4, 3, 2, 1} v\u00e0 b\u1ea1n mu\u1ed1n ch\u1ecdn 4 ph\u1ea7n t\u1eed v\u00e0 s\u1eafp x\u1ebfp ch\u00fang theo m\u1ed9t th\u1ee9 t\u1ef1 c\u1ee5 th\u1ec3, s\u1ed1 ch\u1ec9nh h\u1ee3p c\u1ee7a ch\u00fang l\u00e0: A(6,4) = 6! \/ (6-4)! = (6\u00d75\u00d74\u00d73\u00d72\u00d71) \/ (2×1) = 360. Do \u0111\u00f3, c\u00f3 360 c\u00e1ch ch\u1ecdn v\u00e0 s\u1eafp x\u1ebfp 4 ph\u1ea7n t\u1eed t\u1eeb t\u1eadp h\u1ee3p n\u00e0y.<\/span><\/p>\n

K\u1ebft lu\u1eadn<\/b><\/h2>\n

Nh\u01b0 v\u1eady, Ho\u00e0ng H\u00e0 Mobile \u0111\u00e3 chia s\u1ebb cho b\u1ea1n t\u1ea5t c\u1ea3 th\u1ee7 thu\u1eadt t\u00ednh to\u00e1n <\/span>c\u00f4ng th\u1ee9c ch\u1ec9nh h\u1ee3p<\/b>, t\u1ed5 h\u1ee3p v\u00e0 ho\u00e1n v\u1ecb. C\u00e1c c\u00f4ng th\u1ee9c n\u00e0y kh\u00f4ng qu\u00e1 kh\u00f3 \u0111\u1ec3 \u00e1p d\u1ee5ng, nh\u01b0ng b\u1ea1n c\u1ea7n ph\u1ea3i hi\u1ec3u r\u00f5 \u0111\u01b0\u1ee3c b\u1ea3n ch\u1ea5t, c\u0169ng nh\u01b0 ph\u00e2n bi\u1ec7t \u0111\u01b0\u1ee3c t\u1eebng c\u00f4ng th\u1ee9c kh\u00e1c nhau.<\/span><\/p>\n

Xem th\u00eam:<\/b><\/p>\n