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In-how-many-ways-can-the-letters-of-the-word-LEVITATE-be-arranged-if-the-vowels-must-not-be-next-to-each-other-

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Question Number 118822 by Don08q last updated on 20/Oct/20
In how many ways can the letters of the word LEVITATE be arranged if the vowels must not be next to each other
$$\:\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\: \\ $$$$\:\mathrm{the}\:\mathrm{word}\:{LEVITATE}\:\mathrm{be}\:\mathrm{arranged}\:\mathrm{if} \\ $$$$\:\mathrm{the}\:\mathrm{vowels}\:\mathrm{must}\:\mathrm{not}\:\mathrm{be}\:\mathrm{next}\:\mathrm{to}\:\mathrm{each} \\ $$$$\:\mathrm{other} \\ $$
Answered by benjo_mathlover last updated on 20/Oct/20
vowels : A,I,E ,E □■□■□■□■□ = ((4!)/(2!))×((4!)/(2!))× ((5),(4) ) = 144×5=720 ■=vowels □=consonant
$${vowels}\::\:{A},{I},{E}\:,{E} \\ $$$$\Box\blacksquare\Box\blacksquare\Box\blacksquare\Box\blacksquare\Box\:=\:\frac{\mathrm{4}!}{\mathrm{2}!}×\frac{\mathrm{4}!}{\mathrm{2}!}×\begin{pmatrix}{\mathrm{5}}\\{\mathrm{4}}\end{pmatrix}\:=\:\mathrm{144}×\mathrm{5}=\mathrm{720} \\ $$$$\blacksquare={vowels} \\ $$$$\Box={consonant} \\ $$
Answered by mr W last updated on 20/Oct/20
□■□■□■□■ ■□□■□■□■ ■□■□□■□■ ■□■□■□□■ ■□■□■□■□ ⇒5×((4!)/(2!))×((4!)/(2!))=720
$$\Box\blacksquare\Box\blacksquare\Box\blacksquare\Box\blacksquare \\ $$$$\blacksquare\Box\Box\blacksquare\Box\blacksquare\Box\blacksquare \\ $$$$\blacksquare\Box\blacksquare\Box\Box\blacksquare\Box\blacksquare \\ $$$$\blacksquare\Box\blacksquare\Box\blacksquare\Box\Box\blacksquare \\ $$$$\blacksquare\Box\blacksquare\Box\blacksquare\Box\blacksquare\Box \\ $$$$\Rightarrow\mathrm{5}×\frac{\mathrm{4}!}{\mathrm{2}!}×\frac{\mathrm{4}!}{\mathrm{2}!}=\mathrm{720} \\ $$

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