The-number-of-words-can-be-made-formed-using-all-letters-in-the-word-amutasia-by-not-containing-the-three-vowels-side-by-side-is-

Question Number 77777 by jagoll last updated on 10/Jan/20

T h e n u m b e r

o f w o r d s c a n b e m a d e f o r m e d

u s i n g a l l l e t t e r s i n t h e w o r d

^{'} {a m u t a s i a}^{'} b y n o t c o n t a i n i n g

t h e t h r e e v o w e l s s i d e b y s i d e i s ?

Answered by mr W last updated on 10/Jan/20

A M U T A S I A c o n t a i n s

3 \times A

1 \times I

1 \times U

1 \times M

1 \times S

1 \times T

t o t a l l y 8 l e t t e r s

t o t a l n u m b e r o f w o r d s f o r m e d u s i n g

a l l l e t t e r s :

\frac{8!}{3!} = 6720

a m o n g t h e m n u m b e r o f w o r d s c o n t a i n i n g

t h e t h r e e v o w e l s s i d e b y s i d e :

\frac{3 \times 6! 3!}{2!} = 6480

\Rightarrow r e q u e s t e d n u m b e r : 6720 - 6480 = 240

Commented by jagoll last updated on 10/Jan/20

s i r w h y \frac{3 \times 6! 3!}{2!} s i r ? p l e a s e g i v e e x a m p l e

Commented by mr W last updated on 10/Jan/20

i n f a c t t h e q u e s t i o n i s n o t e x a c t .

t h e t h r e e v o w o l s s h o u l d n o t b e s i d e b y s i d e .

b u t a r e S I A A U M A T a n d S I A A A U M T

v a l i d ? f o r m e t h e y a r e v a l i d . s i d e b y

s i d e m e a n s f o r m e I A U, b u t n o t I A A U .

Commented by jagoll last updated on 10/Jan/20

v a l i d s i r

Commented by mr W last updated on 10/Jan/20

t h e n t h e a n s w e r i s 240 . i s t h i s c o r r e c t ?

Commented by jagoll last updated on 10/Jan/20

t h e p r o b l e m i s n o t m u l t i p l e

c h o i c e s i r . i s t i l l {d o n}^{'} t

u n d e r s t a n d t h e f o r m u l a

r e d u c t i o n \frac{3 \times 6! 3!}{2!}

Commented by jagoll last updated on 10/Jan/20

t h e f i r s t 3 n u m b e r a r e m a n y t y p e s

o f v o w e l s . w h y i s d i v i d e d 2! ?

Commented by malwaan last updated on 10/Jan/20

w i t h r e p e a t i n g l e t t e r s o r n o t ?

Commented by jagoll last updated on 10/Jan/20

n o t s i r

Commented by mr W last updated on 10/Jan/20

t o j a g o l l s i r :

i m a g e y o u m a r k t h e t h r e e l e t t e r s A w i t h

a p e n a s A_{1}, A_{2}, A_{3} a n d t r e a t t h e m a s

d i f f e r e n t l e t t e r s . c e r t a i n l y t h e y a r e

i n f a c t i d e n t i c a l .

t h e n w e s e l e c t t h r e e v o w o l s, e . g . A_{1}, I, U,

a s a g r o u p, s a y [A_{1} I U], a s i f t h e y w e r e

a s i n g l e l e t t e r . t h e n w e a r r a n g e t h e

6 “ {l e t t e r s}^{″} : [A_{1} I U], A_{2}, A_{3}, M, S, T .

t h e r e a r e 6! w a y s t o a r r a n g e t h e m .

b u t A_{2} a n d A_{3} a r e i n f a c t i d e n t i c a l,

t h e r e f o r e t h e r e a r e i n f a c t o n l y

\frac{6!}{2!} d i f f e r e n t w a y s . o n t h e o t h e r s i d e,

w e c a n a r r a n g e t h e t h r e e v o w o l s A_{1}, I, U

i n s i d e t h e g r o u p i n 3! w a y s . t h e r e f o r e w e h a v e

t o t a l l y \frac{6! 3!}{2!} w a y s w h e n w e t a k e A_{1} a s

v o w o l . s i n c e w e c a n a l s o s e l e c t A_{2} o r

A_{3} a s v o w o l, t h e n u m b e r o f a l l p o s s i b i l t i e s

i s \frac{3 \times 6! 3!}{2!} .

c l e a r ?

Commented by jagoll last updated on 10/Jan/20

y e s s i r . t h a n k s y o u v e r y m u c h

Commented by malwaan last updated on 10/Jan/20

g r e a t s i r M r W

t h a n k y o u v e r y m u c h

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