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


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Question Number 124829 by bramlexs22 last updated on 06/Dec/20

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

	$t o a r r a n g e a t f i r s t t h e f o u r l e t t e r s$ $L M T R, t h e r e a r e 4! w a y s .$ $◻ L ◻ M ◻ T ◻ R ◻$ $t o a r r a n g e t h e f o u r l e t t e r s A i n t h e$ $f i v e ◻ p o s i t i o n s, t h e r e a r e C_{4}^{5} w a y s .$ $\Rightarrow t o t a l l y 4! \times C_{4}^{5} = 120$
	Commented by bramlexs22 last updated on 06/Dec/20

	$t h a n k y o u$
	Answered by john_santu last updated on 06/Dec/20

	$(∙) W e f i r s t a r r a n g e t h e 4 A^{'} s i n a r o w$ $o n e w a y :\equiv \underset{1^{s t}}{-} A \underset{2^{n d}}{-} A \underset{3^{r d}}{-} A \underset{4^{t h}}{-} A \underset{5^{t h}}{-}$ $T h e t r e a t 4 l e t t e r s^{'} L, M, T, R^{'} a s i d e n t i c a l$ $^{'} x^{'} . S i n c e n o t w o A^{'} s a r e a d j a c e n t o n e$ $^{'} x, m u s t b e p u t i n t h e 2^{n d}, 3^{r d} a n d 4^{t h} p l a c e s$ $(t h i s c a n b e d o n e i n o n e w a y) \Rightarrow \underset{1^{s t}}{-} A \underset{2^{n d}}{\overset{x}{-}} A \underset{3^{r d}}{\overset{x}{-}} A \underset{4^{t h}}{\overset{x}{-}} A \underset{5^{t h}}{-}$ $N o w t h e r e m a i n i n g 1^{'} x^{'} c a n b e p u t i n t h e$ $5 p l a c e s a r b i t r a r i l y i n (\begin{matrix} 1 + 5 - 1 \\ 1 \end{matrix}) = (\begin{matrix} 5 \\ 1 \end{matrix}) = 5$ $T h u s t h e d e s i r e d n u m b e r o f w a y s i s g i v e n$ $b y 5 \times 4! = 5 \times 24 = 120$
	Commented by bramlexs22 last updated on 06/Dec/20

	$t h a n k y o u$
	Commented by liberty last updated on 06/Dec/20
	it's a good explanation to students so they can easily understand it
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