How many ways are there to arrange the letters of the word ′ VISITING′ if no two I′s are adjacent ?


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Question Number 124826 by bramlexs22 last updated on 06/Dec/20

$H o w m a n y w a y s a r e t h e r e t o a r r a n g e$ $t h e l e t t e r s o f t h e w o r d^{'} {V I S I T I N G}^{'}$ $i f n o t w o I^{'} s a r e a d j a c e n t ?$
	Answered by mr W last updated on 06/Dec/20

	$M e t h o d I$ $a r r a n g e a t f i r s t t h e f i v e o t h e r l e t t e r s$ $a n d t h e n t h e t h r e e I^{'} s i n 6 p o s i t i o n s :$ $◻ V ◻ S ◻ T ◻ N ◻ G ◻$ $5! \times C_{3}^{6} = 2400$ $M e t h o d I I$ $a r r a n g e a t f i r s t t h e t h r e e I^{'} s a n d$ $t h e n t h e f i v e o t h e r l e t t e r s :$ $◻ I ◼ I ◼ I ◻$ $◻ = z e r o o r m o r e o t h e r l e t t e r s$ $◼ = o n e o r m o r e o t h e r l e t t e r s$ ${(\underset{◻}{1 + x + x^{2} + . . .})}^{2} {(\underset{◼}{x + x^{2} + x^{3} + . . .})}^{2}$ $= \frac{x^{2}}{{(1 - x)}^{4}}$ $= x^{2} \underset{k = 0}{\sum^{\infty}} C_{3}^{k + 3} x^{k}$ $c o e f . o f t e r m x^{5} :$ $k = 3 \Rightarrow C_{3}^{6}$ $\Rightarrow C_{3}^{6} \times 5! = 2400$
	Commented by john_santu last updated on 06/Dec/20

	$g r e a t m e t h o d I I$
	Commented by malwan last updated on 06/Dec/20

	$I w a n t t o m a s t e r m e t h o d I I$ $l i k e y o u m r W$ $t h a n k y o u$
	Commented by mr W last updated on 06/Dec/20

	${i t}^{'} s a v e r y p o w e r f u l m e t h o d, t r y t o$ $a p p l y i t!$
	Answered by john_santu last updated on 06/Dec/20

	Commented by bramlexs22 last updated on 06/Dec/20

	$g r e a t!$
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