Misali na Tambayoyin Tattaunawa Masu Canzawa
Canzawa shine sake fasalin wani tsari ko abubuwa a cikin wani tsari na musamman. A cikin lissafi, ana amfani da wannan ra'ayi akai-akai don ƙididdige hanyoyin da za a iya tsara rukunin abubuwa. A ƙasa, za mu tattauna misalai da yawa na matsalolin canjin yanayi da cikakkun bayanai game da su.
Ma'anar Canzawa
Canzawa na saitin wani tsari ne na sake fasalin abubuwan da ke cikinsa a cikin wani tsari na musamman. Idan akwai abubuwa masu \(n\), ana nuna canjin ta hanyar \(P(n) \) ko musamman, \(P(n, r) \) don \(r \) canjin abubuwa masu \(n \) . Tsarin asali na canjin shine:
\[ P(n) = n! \]
inda \(n! \) (n factorial) samfurin dukkan lambobi masu kyau ƙasa da ko daidai da \(n \).
A halin yanzu, dabarar canza abubuwa \( r \) na abubuwa \( n \) shine:
\[ P(n, r) = \frac{n!}{(nr)!} \]
Tambayoyi da Tattaunawa Samfura
Misali Tambaya ta 1
matsala:
Hanyoyi nawa za a iya shirya littattafai guda 4 daban-daban a kan shiryayye?
Tattaunawa:
Don shirya littattafai 4 daban-daban, za mu iya amfani da dabarar canza littattafai don ƙididdige duk shirye-shiryen littattafan da za a iya yi:
\[ P(4) = 4! = 4 \sau 3 \sau 2 \sau 1 = 24 \]
Don haka, akwai hanyoyi 24 don shirya littattafai 4 daban-daban a kan shiryayye.
Misali Tambaya ta 2
matsala:
Hanyoyi nawa ne ake da su don zaɓar da kuma shirya mambobi 3 na ƙungiyar mambobi 5 a cikin tsari da aka bayar?
Tattaunawa:
Muna amfani da dabarar canza launi \( P(n, r) \) inda \(n = 5 \) da \( r = 3 \):
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \sau 4 \sau 3 \sau 2!}{2!} = 5 \sau 4 \sau 3 = 60 \]
Don haka, akwai hanyoyi 60 don zaɓar da kuma shirya mambobi 3 na ƙungiyar mambobi 5 a cikin wani tsari.
Misali Tambaya ta 3
matsala:
Hanyoyi nawa za a iya tsara kalmar "MATS" don kada a maimaita haruffa?
Tattaunawa:
Kalmar "MATH" ta ƙunshi haruffa huɗu daban-daban. Za mu iya amfani da dabarar canza abubuwa don ƙididdige duk shirye-shiryen waɗannan haruffa:
\[ P(4) = 4! = 4 \sau 3 \sau 2 \sau 1 = 24 \]
Don haka, akwai hanyoyi 24 don tsara haruffan da ke cikin kalmar "MATS".
Misali Tambaya ta 4
matsala:
Daga lambobi 1, 2, 3, 4, 5, lambobi nawa ne za a iya samu idan ba a maimaita lambobi ba?
Tattaunawa:
Domin samar da lamba mai lambobi 3 daga lambobi 5 daban-daban inda babu lamba da aka maimaita, muna amfani da canjin \( P(5, 3) \):
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \sau 4 \sau 3 \sau 2!}{2!} = 5 \sau 4 \sau 3 = 60 \]
Don haka, akwai hanyoyi 60 don samar da lamba mai lambobi 3 daga lambobi 1, 2, 3, 4, da 5 ba tare da maimaita kowace lamba ba.
Misali Tambaya ta 5
matsala:
Akwai 'yan wasa 6, A, B, C, D, E, da F. Za a shirya su a cikin manyan 'yan wasa 3 don wasan. Hanyoyi nawa za a iya shirya 'yan wasa uku?
Tattaunawa:
A nan an nemi mu shirya 'yan wasa 3 a cikin takamaiman tsari daga cikin jimillar 'yan wasa 6. Tsarin da aka yi amfani da shi shine canjin \( P(n, r) \) inda \( n = 6 \) da \( r = 3 \):
\[ P(6, 3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} = \frac{6 \sau 5 \sau 4 \sau 3!}{3!} = 6 \sau 5 \sau 4 = 120 \]
Don haka, akwai hanyoyi 120 don shirya 'yan wasa 3 daga cikin 6 a cikin wani tsari na musamman.
Misali Tambaya ta 6
matsala:
Ka tantance adadin canje-canjen da ke cikin kalmar "JAAMI'A" don haka wasalan koyaushe suna kusa da juna.
Tattaunawa:
Kalmar "JAM'I" ta ƙunshi haruffa 11, kuma wasulan sune U, I, E, I, A. Yi la'akari da wannan rukunin wasulan a matsayin raka'a ɗaya.
Don haka, muna da: (UIEIA), N, V, R, S, T, da S (an yi la'akari da su a matsayin raka'a ɗaya). Sannan dole ne mu shirya waɗannan raka'a 7:
\[ P(7) = 7! = 5040 \]
Duk da haka, a cikin ƙungiyar mawaƙa (UIEIA), ana iya shirya su a cikin:
\[ P(5) = 5! = 120 \]
Saboda haka, jimlar canje-canje sune:
\[ 7! \sau 5! = 5040 \sau 120 = 604800 \]
Don haka, akwai hanyoyi 604800 don samar da kalmar "JAAMI'A" inda duk wasulan ke kusa da juna koyaushe.
Kammalawa
Canzawa shine tsarin abubuwa ko saiti a cikin takamaiman tsari, kuma wannan ra'ayi yana da aikace-aikace da yawa a fannoni daban-daban, gami da lissafi, kimiyyar kwamfuta, da ƙididdiga. Ta hanyar gano da aiwatar da dabarar da ta dace, za mu iya ƙididdige adadin shirye-shiryen da za a iya yi cikin sauƙi.
Misalan da aka bayar sun nuna yadda dabarun canza ma'auni ke aiki da kuma yadda za a iya amfani da su a yanayi daban-daban. Fahimtar ma'auni yana da mahimmanci don magance matsalolin haɗaka masu rikitarwa kuma yana da matuƙar amfani wajen haɓaka dabarun warware matsaloli.
