$The probability that a randomly chosen 5 digit number is made from exactly two digits is$

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$\frac{135}{10^{4}}$

$\frac{150}{10^{4}}$

$\frac{121}{10^{4}}$

$\frac{134}{10^{4}}$

answer is A.

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Detailed Solution

${The number of five digit numbers when the repition allowed is 9·10}^{4} T h e n u m b e r o f f i v e d i g i t n u m b e r s w i t h t h e s p e i c i f i e d t w o d i g i t s T h e r e a r e t w o c a s e s . C a s e (i) : S u p p o s e t h a t t h e n u m b e r h a s n o z e r o . s e l e c t t w o d i g i t s f r o m 9 a n d a r r a n g e t h e m i n f i v e b \ln a k s w i t h r e p e t i o n . t h i s c a n b e d o n e i n C (9, 2) (2^{5} - 2) C a s e (i i) S u p p o s e t h a t t h e n u m b e r h a s z e r o S e l e c t o n e d i g i t f r o m 9, t h i s c a n b e d o n e i n 9 w a y s a n d f i x i t i n t h e f i r s t p l a c e r e m a i n i n g 4 d i g i t s c a n b e f i l l e d w i t h a n y o f t h e s e t w o d i g i t s t h i s c a n b e d o n e i n 9 \cdot (2^{4} - 1) (t h e n u m b e r f o r m e d b y o n l y o n e d i g i t s u b t r a c t e d f r o m t e b u n c h) T h e r e q u i r e d p r o b a b i l i t y i s \frac{C (9, 2) (2^{5} - 2) + C (9, 1) (2^{4} - 1)}{9 \cdot 10^{4}} = \frac{135}{10^{4}}$