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-

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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