how many different words can be formed from the letters in aaacdefgbbbb such that a “a” and a “b” are not next to each other? (see also Q#195606)


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Question Number 195672 by mr W last updated on 07/Aug/23

$h o w m a n y d i f f e r e n t w o r d s c a n b e$ $f o r m e d f r o m t h e l e t t e r s i n$ $a a a c d e f g b b b b$ $s u c h t h a t a ‘ ‘ a^{″} a n d a ‘ ‘ b^{″} a r e n o t$ $n e x t t o e a c h o t h e r ?$ $You can't use 'macro parameter character #' in math mode$
	Answered by liuxinnan last updated on 10/Aug/23

	Commented by mr W last updated on 10/Aug/23

	$‘ ‘ a^{″} a n d ‘ ‘ b^{″} s h o u l d n o t b e n e x t t o$ $e a c h o t h e r .$ $t w o o r m o r e ‘ ‘ a^{″} c a n b e t o g e t h e r .$ $t w o o r m o r e ‘ ‘ b^{″} c a n b e t o g e t h e r .$ $t w o o r m o r e o t h e r l e t t e r s c a n b e t o g e t h e r .$ $a a c a d e b b b f b g i f e . g . v a l i d .$
	Commented by liuxinnan last updated on 10/Aug/23

	$I m i s u n d e r s t o o d i t$
	Commented by liuxinnan last updated on 10/Aug/23

	Commented by mr W last updated on 11/Aug/23

	$t h a n k y o u f o r t r y i n g!$ $i n e a c h ⊝ y o u c a n p u t n o t h i n g (e m p t y)$ $o r p u t o n e t i l l n ‘ ‘ a^{″}$ $o r p u t o n e t i l l m ‘ ‘ b^{″}$ $i n a l l ⊝ y o u s h o u l d t o t a l l y h a v e n ‘ ‘ a^{″}$ $a n d m ‘ ‘ b^{″} .$ $y o u s e e, i t i s n o t s o a n e a s y q u e s t i o n$ $a s i t l o o k s l i k e .$
	Answered by mr W last updated on 11/Aug/23

	$w e a r r a n g e a t f i r s t t h e 5 o t h e r l e t t e r s .$ $t h e r e a r e 5! w a y s t o d o t h i s .$ $◻ c ◻ d ◻ e ◻ f ◻ g ◻$ $i n e a c h o f t h e 6 b o x e s (◻) w e c a n p u t$ $n o t h i n g (e m p t y) \Leftrightarrow w e a d d 0 t o i t$ $o r ‘ ‘ a^{″} \Leftrightarrow w e a d d 1 t o i t$ $o r ‘ ‘ {a a}^{″} \Leftrightarrow w e a d d 2 t o i t$ $o r ‘ ‘ {a a a}^{″} \Leftrightarrow w e a d d 3 t o i t$ $o r ‘ ‘ {a a a a}^{″} \Leftrightarrow w e a d d 4 t o i t$ $o r ‘ ‘ b^{″} \Leftrightarrow w e a d d 100 t o i t$ $o r ‘ ‘ {b b}^{″} \Leftrightarrow w e a d d 200 t o i t$ $o r ‘ ‘ {b b b}^{″} \Leftrightarrow w e a d d 300 t o i t$ $t o t a l l y w e s h o u l d p u t 4 ‘ ‘ a^{″} a n d 3 ‘ ‘ b^{″}$ $i n t h e 6 b o x e s, t h a t m e a n s t h e s u m$ $o f a l l 6 b o x e s s h o u l d b e 4 + 300 = 304 .$ $s o t h e n u m b e r o f w a y s t o d o t h i s i s$ $t h e c o e f f i c i e n t o f t e r m x^{304} i n t h e$ $e x p a n s i o n o f g e n e r a t i n g f u n c t i o n$ ${(1 + x + x^{2} + x^{3} + x^{4} + x^{100} + x^{200} + x^{300})}^{6},$ $w h i c h i s 1770 .$ $h e n c e t h e n u m b e r o f v a l i d w o r d s w e$ $c a n f o r m i s 1770 \times 5! = 212 400 ✓$
	Commented by mr W last updated on 11/Aug/23

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