Tusaalaha Su'aalaha Wadahadalka Isku-dhafka ah
Isku-darka waa fikrad muhiim ah oo ku saabsan aragtida iyo tirakoobka suurtogalnimada, taasoo noo oggolaanaysa inaan tirinno tirada siyaabaha lagu xulan karo walxaha set iyada oo aan loo eegin nidaamka. Isku-darka, kala-horeynta xulashada isma beddelin, si ka duwan kala-goynta, halkaas oo kala-horeynta xulashada ay muhiim tahay. Isku-darka waxaa loo isticmaalaa meelo kala duwan, laga bilaabo sayniska kombiyuutarka iyo bayoolajiga ilaa dhaqaalaha iyo xisaabta. Maqaalkani wuxuu sharxi doonaa fikradda aasaasiga ah ee isku-darka wuxuuna bixin doonaa dhowr dhibaato iyo doodo tusaale ah si uu kaaga caawiyo fahamkaaga.
Fikradda Aasaasiga ah ee Isku-darka
Kahor inta aynaan u gudbin dhibaatooyinka tusaalaha ah, aan ku bilowno faham aasaasi ah oo ku saabsan isku-darka.
Isku darka walxaha \(n\) ee la qaaday \(r\) markiiba waxaa lagu tilmaamayaa calaamadda \(\binom{n}{r}\) ama \(\mathbf{C}(n, r)\). Qaacidada lagu xisaabinayo isku-darka waa:
\[ \binom{n}{r} = \frac{n!}{r!(nr)!} \]
– \( n \) waa tirada guud ee walxaha la heli karo.
– \( r \) waa tirada walxaha la dooranayo.
– ! (factorial) waa hawlgal xisaabeed oo tiro ku dhufta dhammaan tirooyinka togan ee ka hooseeya ilaa 1.
Su'aalo iyo Doodo Tusaale ah
Su'aal Tusaale ah 1: Xulashada Koox ka socota Koox Arday ah
Su'aal:
Waxaa fasalka ku jira 10 arday. Immisa siyaabood ayaa loo dooran karaa 4 arday oo fasalka ka tirsan si ay uga qayb qaataan tartanka?
Dood:
Isticmaalka qaacidada isku-darka:
\[ \binom{10}{4} = \frac{10!}{4! \cdot (10-4)!} = \frac{10!}{4! \cdot 6!} \]
Xisaabinta factorialka:
– \( 10! = 10 \jeer 9 \jeer 8 \jeer 7 \jeer 6! \)
– \( 4! = 4 \jeer 3 \jeer 2 \jeer 1 = 24 \)
– \( 6! \) horeyba ayuu ugu sarreeyaa, sidaas darteed wuxuu tirtiri karaa 6! ee \(10!\).
Markaa:
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]
Markaa, waxaa jira 210 siyaabood oo lagu dooran karo 4 arday 10 arday.
Su'aal Tusaale 2: Isku-darka Isticmaalka Qalabka Kaararka
Su'aal:
Laga soo bilaabo sagxad ka kooban 52 kaar, imisa siyaabood ayaa loo dooran karaa gacan 5-kaar ah oo turubka ah?
Dood:
Isticmaalka qaacidada isku-darka:
\[ \binom{52}{5} = \frac{52!}{5! \cdot (52-5)!} = \frac{52!}{5! \cdot 47!} \]
Xisaabinta factorialka:
– \( 52! = 52 \jeer 51 \jeer 50 \jeer 49 \jeer 48 \jeer 47! \)
– \( 5! = 5 \jeer 4 \jeer 3 \jeer 2 \jeer 1 = 120 \)
– \( 47! \) horeyba ayuu ugu sarreeyaa, sidaas darteed wuxuu tirtiri karaa 47! ee \(52!\).
Markaa:
\[ \binom{52}{5} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = \frac{311875200}{120} = 2598960 \]
Marka, waxaa jira 2,598,960 siyaabood oo lagu dooran karo 5 kaar oo ka mid ah 52 kaar.
Su'aal Tusaale ah 3: Xisaabinta Isku-darka Xulashada Kooxda Ciyaaraha
Su'aal:
Koox kubbadda kolayga ah waxay ka kooban tahay 15 ciyaartoy. Haddii tababaruhu doonayo inuu doorto 5 ciyaartoy oo bilow ah, imisa siyaabood ayaa loo samayn karaa tan?
Dood:
Isticmaalka qaacidada isku-darka:
\[ \binom{15}{5} = \frac{15!}{5! \cdot (15-5)!} = \frac{15!}{5! \cdot 10!} \]
Xisaabinta factorialka:
– \( 15! = 15 \jeer 14 \jeer 13 \jeer 12 \jeer 11 \jeer 10! \)
– \( 5! = 5 \jeer 4 \jeer 3 \jeer 2 \jeer 1 = 120 \)
– \( 10! \) horeyba ayuu ugu sarreeyaa, sidaas darteed wuxuu tirtiri karaa 10! ee \(15!\).
Markaa:
\[ \binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = \frac{360360}{120} = 3003 \]
Markaa, waxaa jira 3,003 siyaabood oo lagu dooran karo 5 ciyaartoy oo ka mid ah 15 ciyaartoy.
Su'aal Tusaale ah 4: Isku-darka Bayoolaji
Su'aal:
Beerta dhexdeeda, waxaa ku yaal 8 nooc oo ubax ah. Immisa siyaabood ayaa loo dooran karaa 3 nooc oo ubax ah si loo sameeyo habaynta ubaxa?
Dood:
Isticmaalka qaacidada isku-darka:
\[ \binom{8}{3} = \frac{8!}{3! \cdot (8-3)!} = \frac{8!}{3! \cdot 5!} \]
Xisaabinta factorialka:
– \( 8! = 8 \jeer 7 \jeer 6 \jeer 5! \)
– \( 3! = 3 \jeer 2 \jeer 1 = 6 \)
– \( 5! \) horeyba ayuu ugu sarreeyaa, sidaas darteed wuxuu tirtiri karaa 5! ee \(8!\).
Markaa:
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]
Markaa, waxaa jira 56 siyaabood oo lagu dooran karo 3 nooc oo ubax ah 8 nooc oo ubax ah.
Su'aal Tusaale ah 5: Isku-darka Dhismaha Kooxda
Su'aal:
Waxaad haysataa 12 buug oo kala duwan oo ku yaal khaanadaha buugaagta. Immisa siyaabood ayaad u dooran kartaa 5 buug oo ka mid ah 12-ka?
Dood:
Isticmaalka qaacidada isku-darka:
\[ \binom{12}{5} = \frac{12!}{5! \cdot (12-5)!} = \frac{12!}{5! \cdot 7!} \]
Xisaabinta factorialka:
– \( 12! = 12 \jeer 11 \jeer 10 \jeer 9 \jeer 8 \jeer 7! \)
– \( 5! = 5 \jeer 4 \jeer 3 \jeer 2 \jeer 1 = 120 \)
– \( 7! \) horeyba ayuu ugu sarreeyaa, sidaas darteed wuxuu tirtiri karaa 7! ee \(12!\).
Markaa:
\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} = \frac{95040}{120} = 792 \]
Markaa, waxaa jira 792 siyaabood oo lagu dooran karo 5 buug laga bilaabo 12 buug.
Gabagabo
Isku-darka waa fikrad aad waxtar u leh dhinacyo kala duwan, gaar ahaan marka aan u baahannahay inaan ogaano tirada siyaabaha loo doorto walxaha iyada oo aan loo eegin kala horreynta. Annagoo adeegsanayna qaacidada isku-darka aasaasiga ah \(\binom{n}{r}\), waxaan si dhakhso leh oo fudud u xisaabin karnaa, ilaa inta aan fahanno sida factorials-ku u shaqeeyaan. Iyada oo loo marayo tusaalooyinka kor ku xusan, waxaan rajeyneynaa inaan yeelanno faham dhammaystiran oo cad oo ku saabsan isku-darka. Sii wad ku celcelinta dhibaatooyin kala duwan si aad u sii xoojiso fahamkaaga.
