The number of ways arrangements of the word ′MASKARA′ with exactly 2 A′s are adjacent??


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Question Number 124916 by liberty last updated on 07/Dec/20

$T h e n u m b e r o f w a y s a r r a n g e m e n t s$ $o f t h e w o r d^{'} {M A S K A R A}^{'} w i t h e x a c t l y$ $2 A^{'} s a r e a d j a c e n t ? ?$
	Answered by mr W last updated on 07/Dec/20

	$_M_S_K_R_$ $4! \times P_{2}^{5} = 480$
	Commented by mr W last updated on 07/Dec/20

	$M e t h o d I I$ $3 A^{'} s a d j a c e n t : 5!$ $a t l e a s t 2 A^{'} s a d j a c e n t : 6!$ $e x a c t l y 2 A^{'} s a d j a c e n t : 6! - 2 \times 5! = 480$
	Commented by bemath last updated on 07/Dec/20

	$i g o t 960 s i r . w h a t w r o n g ?$
	Commented by mr W last updated on 07/Dec/20

	$h o w d i d y o u g e t ?$
	Answered by liberty last updated on 07/Dec/20

	$(∙) \underset{1^{s t}}{-} \overset{A A}{-} \underset{2^{n d}}{-} \overset{A}{-} \underset{3^{r d}}{-}$ $T r e a t 4 l e t t e r s M, S, K, R a s i d e n t i c a l^{'} x^{'}$ $s i n c e e x a c t l y 2 A^{'} s a r e a d j a c e n t, o n e x m u s t b e$ $p u t i n 2^{n d} p l a c e s \Rightarrow \underset{1^{s t}}{-} A A \underset{2^{n d}}{\overset{x}{-}} A \underset{3^{r x}}{-}$ $t h e r e m a i n i n g 3 x^{'} s c a n b e p u t i n (\begin{matrix} 3 + 3 - 1 \\ 3 \end{matrix}) = (\begin{matrix} 5 \\ 3 \end{matrix}) = 10$ $(∙ ∙) \underset{1^{s t}}{-} A \underset{2^{n d}}{-} A A \underset{3^{r d}}{-} s i m i l a r t o c a s e s (∙)$ $t h e r e f o r e t h e d e s i r e d n u m b e r o f w a y s$ $i s g i v e n b y 2 \times (\begin{matrix} 5 \\ 3 \end{matrix}) \times 4! = 20 \times 24 = 480$
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