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

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

H o w m a n y w a y s t o a r r a n g e l e t t e r s

u s i n g l e t t e r s i n t h e w o r d O K I N K O

(a) 1260 (b) 1160 (c) 980

(d) 880 (e) 720

Commented by liberty last updated on 12/Nov/20

case (1) : 1 - alphabet \Rightarrow C_{1}^{4} = 4

case (2) : 2 - alphabet \to {\begin{cases} OO, KK = 2 \\ OK, KO = 2 \\ OI, ON, KI, KN, IN = 2! \times 5 = 10 \end{cases}

case (3) : 3 - aphabet

\to {\begin{cases} OOK = \frac{3!}{2!} = 3, OKK = \frac{3!}{2} = 3 \\ OO \to {\begin{cases} I \\ N \end{cases} = 2 \times \frac{3!}{2!} = 6; KK \to {\begin{cases} I \\ N \end{cases} = 2 \times \frac{3!}{2!} = 6 \\ OK \to {\begin{cases} I \\ N \end{cases} = 2 \times 3! = 12 \end{cases}

case (4) : 4 - alphabet

\to {\begin{cases} OOKK = \frac{4!}{2! . 2!} = 6 \\ OOK {\begin{cases} I \\ N \end{cases} = 2 \times \frac{4!}{2!} = 24 \\ OKK {\begin{cases} I \\ N \end{cases} = 2 \times \frac{4!}{2!} = 24 \\ OKIN = 4! = 24 \end{cases}

case (5) : 5 - alphabet

\to {\begin{cases} OOKK {\begin{cases} I \\ N \end{cases} = 2 \times \frac{5!}{2! . 2!} = 60 \\ OKKNI = \frac{5!}{2!} = 60 \\ OOKNI = \frac{5!}{2!} = 60 \end{cases}

case (6) : 6 - alphabet = \frac{6!}{2! . 2!} = 180

Totally = 18 + 30 + 78 + 180 + 180

360 + 108 + 18

486.0

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