Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 121851 by bemath last updated on 12/Nov/20

How many ways to arrange letters using letters in the word OKINKO (a) 1260 (b) 1160 (c) 980 (d) 880 (e) 720

$${How}\:{many}\:{ways}\:{to}\:{arrange}\:{letters}\: \\ $$$${using}\:{letters}\:{in}\:{the}\:{word}\:{OKINKO} \\ $$$$\left({a}\right)\:\mathrm{1260}\:\:\:\:\:\left({b}\right)\:\mathrm{1160}\:\:\:\:\left({c}\right)\:\mathrm{980} \\ $$$$\left({d}\right)\:\mathrm{880}\:\:\:\:\:\:\:\:\left({e}\right)\:\mathrm{720} \\ $$

Commented by liberty last updated on 12/Nov/20

case(1) :1−alphabet ⇒C_1 ^4 = 4 case(2): 2−alphabet → { ((OO,KK = 2)),((OK,KO=2 )),((OI,ON,KI,KN,IN=2!×5=10)) :} case(3):3−aphabet → { ((OOK=((3!)/(2!)) = 3 ,OKK=((3!)/2)=3)),((OO→ { (I),(N) :} = 2×((3!)/(2!)) = 6 ; KK→ { (I),(N) :}= 2×((3!)/(2!))=6 )),((OK→ { (I),(N) :}=2×3!=12 )) :} case(4): 4−alphabet → { ((OOKK = ((4!)/(2!.2!)) = 6 )),((OOK { (I),(N) :} = 2×((4!)/(2!)) = 24 )),((OKK { (I),(N) :}= 2×((4!)/(2!)) = 24 )),((OKIN = 4! = 24 )) :} case(5):5−alphabet → { ((OOKK { (I),(N) :}=2×((5!)/(2!.2!)) = 60)),((OKKNI = ((5!)/(2!)) = 60 )),((OOKNI = ((5!)/(2!)) = 60 )) :} case(6) : 6−alphabet = ((6!)/(2!.2!)) = 180 Totally = 18+30+78+180+180 360+108+18 486.0

$$\mathrm{case}\left(\mathrm{1}\right)\::\mathrm{1}−\mathrm{alphabet}\:\Rightarrow\mathrm{C}_{\mathrm{1}} ^{\mathrm{4}} \:=\:\mathrm{4} \\ $$$$\mathrm{case}\left(\mathrm{2}\right):\:\mathrm{2}−\mathrm{alphabet}\:\rightarrow\begin{cases}{\mathrm{OO},\mathrm{KK}\:=\:\mathrm{2}}\\{\mathrm{OK},\mathrm{KO}=\mathrm{2}\:}\\{\mathrm{OI},\mathrm{ON},\mathrm{KI},\mathrm{KN},\mathrm{IN}=\mathrm{2}!×\mathrm{5}=\mathrm{10}}\end{cases} \\ $$$$\mathrm{case}\left(\mathrm{3}\right):\mathrm{3}−\mathrm{aphabet} \\ $$$$\rightarrow\begin{cases}{\mathrm{OOK}=\frac{\mathrm{3}!}{\mathrm{2}!}\:=\:\mathrm{3}\:,\mathrm{OKK}=\frac{\mathrm{3}!}{\mathrm{2}}=\mathrm{3}}\\{\mathrm{OO}\rightarrow\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}\:=\:\mathrm{2}×\frac{\mathrm{3}!}{\mathrm{2}!}\:=\:\mathrm{6}\:;\:\mathrm{KK}\rightarrow\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}=\:\mathrm{2}×\frac{\mathrm{3}!}{\mathrm{2}!}=\mathrm{6}\:}\\{\mathrm{OK}\rightarrow\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}=\mathrm{2}×\mathrm{3}!=\mathrm{12}\:}\end{cases} \\ $$$$\mathrm{case}\left(\mathrm{4}\right):\:\mathrm{4}−\mathrm{alphabet} \\ $$$$\rightarrow\begin{cases}{\mathrm{OOKK}\:=\:\frac{\mathrm{4}!}{\mathrm{2}!.\mathrm{2}!}\:=\:\mathrm{6}\:}\\{\mathrm{OOK\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}}\:=\:\mathrm{2}×\frac{\mathrm{4}!}{\mathrm{2}!}\:=\:\mathrm{24}\:}\\{\mathrm{OKK\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}}=\:\mathrm{2}×\frac{\mathrm{4}!}{\mathrm{2}!}\:=\:\mathrm{24}\:}\\{\mathrm{OKIN}\:=\:\mathrm{4}!\:=\:\mathrm{24}\:}\end{cases} \\ $$$$\mathrm{case}\left(\mathrm{5}\right):\mathrm{5}−\mathrm{alphabet} \\ $$$$\rightarrow\begin{cases}{\mathrm{OOKK\begin{cases}{\mathrm{I}}\\{\mathrm{N}}\end{cases}}=\mathrm{2}×\frac{\mathrm{5}!}{\mathrm{2}!.\mathrm{2}!}\:=\:\mathrm{60}}\\{\mathrm{OKKNI}\:=\:\frac{\mathrm{5}!}{\mathrm{2}!}\:=\:\mathrm{60}\:}\\{\mathrm{OOKNI}\:=\:\frac{\mathrm{5}!}{\mathrm{2}!}\:=\:\mathrm{60}\:}\end{cases} \\ $$$$\mathrm{case}\left(\mathrm{6}\right)\::\:\mathrm{6}−\mathrm{alphabet}\:=\:\frac{\mathrm{6}!}{\mathrm{2}!.\mathrm{2}!}\:=\:\mathrm{180} \\ $$$$\mathrm{Totally}\:=\:\mathrm{18}+\mathrm{30}+\mathrm{78}+\mathrm{180}+\mathrm{180} \\ $$$$\mathrm{360}+\mathrm{108}+\mathrm{18} \\ $$$$\mathrm{486}.\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com