How many words with or without meaning each with 2 vowels and 3 consonants can be formed from the letters of the word FRACTION?
 
Find the number of arrangements of the letters of the word INTERMEDIATE
In how many of these arrangements
i) do the words start and end with the letter I
ii) do all the E's always occur together

Asked by ahuja8087 | 8th Jun, 2016, 12:35: AM

Expert Answer:

left parenthesis 1 right parenthesis
T h e r e space a r e space 5 space c o n s o n a n t s comma space F comma space R comma space C comma space T space a n d space N
A n d space t h r e e space v o w e l s comma space A comma space I comma space O
O u t space o f space 5 space c o n s o n a n t s space w e space n e e d space t o space s e l e c t space 3 space a n d space t h i s space c a n space b e space d o n e space i n space C presuperscript 5 subscript 3 space w a y s
O u t space o f space 3 space v o w e l s space w e space n e e d space t o space s e l e c t space 2 space a n d space t h i s space c a n space b e space d o n e space i n space C presuperscript 3 subscript 2 space w a y s
T h u s space comma space t h e r e space a r e space C presuperscript 5 subscript 3 cross times C presuperscript 3 subscript 2 space n u m b e r space o f space w a y s space i n space s e l e c t i n g space 3 space c o n s o n a n t s space a n d space 2 space v o w e l s.
S i n c e space i n space a space w o r d space t h e r e space a r e space 3 space c o n s o n a n t s space a n d space 2 space v o w e l s comma space t h e r e space 5 space l e t t e r s space i n space a space g r o u p comma space w h i c h space t h e m s e l v e s space c a n
b e space a r r a n g e d space i n space 5 factorial space w a y s.
T h u s comma space i n space t o t a l space t h e r e space a r e space 5 factorial cross times open parentheses C presuperscript 5 subscript 3 cross times C presuperscript 3 subscript 2 close parentheses space n u m b e r space o f space w a y s space i n space s e l e c t i n g space 3 space c o n s o n a n t s space a n d space 2 space v o w e l s.

open parentheses 2 close parentheses
C o n s i d e r space t h e space w o r d space I N T E R M E D I A T E
T h e r e space a r e space 12 space l e t t e r s space i n space t o t a l space o u t space o f space w h i c h space E space i s space r e p e a t i n g space 3 space t i m e s comma space I space i s space r e p e a t i n g space 2 space t i m e s space a n d space T space i s space r e p e a t i n g space 2 space t i m e s

left parenthesis i right parenthesis A c c o r d i n g space t h e space q u e s t i o n space s t a t e m e n t comma space p o s i t i o n s space o f space 2 space I s space a r e space f i x e d.
T h u s space t h e r e space a r e space 10 space l e t t e r s space o u t space o f space w h i c h space E space i s space r e p e a t i n g space 3 space t i m e s space a n d space T space i s space r e p e a t i n g space 2 space t i m e s
T h u s comma space t o t a l space n u m b e r space o f space a r r a n g e m e n t s space equals fraction numerator 10 factorial over denominator 3 space factorial cross times 2 space factorial end fraction

left parenthesis i i right parenthesis A c c o r d i n g space t o space t h e space q u e s t i o n space s t a t e m e n t comma space a l l space t h e space t h r e e space E s space s h o u l d space b e space t o g e t h e r.
T h u s comma space apostrophe E E E apostrophe space c a n space b e space c o n s i d e r e d space a s space o n e space u n i t.
T h u s space t h e space l e t t e r s space a r e space I comma space N comma space T comma space R comma space space M comma space D comma space I space A comma space T comma space apostrophe E E E apostrophe
T h e r e space a r e space 10 space l e t t e r s space o u t space o f space w h i c h space space I space i s space r e p e a t i n g space 2 space t i m e s space a n d space T space i s space r e p e a t i n g space 2 space t i m e s
T h u s comma space t o t a l space n u m b e r space o f space a r r a n g e m e n t s space equals fraction numerator 10 factorial over denominator 2 space factorial cross times 2 space factorial end fraction

Answered by Vimala | 8th Jun, 2016, 10:13: AM