Find the number of arrangements that can be formed by using all the letters of the word MATRIX, so that the vowels occupy even places.

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Solution

$N u m b e r o f l e t t e r s i n M A T R I X = 6 e v e n p o s i t i o n s = 2 n d, 4 t h a n d 6 t h t w o v o w e l s A, I s o t w o v o w e l s i n t h r e e p l a c e s c a n b e a r r a n g e d i n = 3 \times 2 = 6 w a y s R e s t f o u r p o s i t i o n s c a n b e f i l l e d b y 4 l e t t e r s i n = 4! w a y s = 24 w a y s ∴ T o t a l p e r m u t a t i o n = 24 \times 6 144$

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