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The number of rational numbers lying in the interval  (2015,2016) all whose digits after the decimal point are non-zero and are in decreasing order is

Option: 1

\mathrm{\sum_{i=1}^{9}{ }^{9} P_{i}}


Option: 2

\mathrm{\sum_{i=1}^{10} 9 P_{i}}


Option: 3

2^{9}-1


Option: 4

2^{10}-1


Answers (1)

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A rational number of the desired category is of the form 2015. \mathrm{x_{1} x_{2} \ldots x_{k}\: where \: 1 \leq k \leq 9\: and \: 9 \geq x_{1}>x_{2}>\ldots>x_{k} \geq 1}. We can choose k digits out 9 in \mathrm{{ }^{9} C_{k}}  ways and arrange them in decreasing order in just one way. Thus, the desired number of rational numbers is \mathrm{{ }^{9} C_{1}+{ }^{9} C_{2}+\ldots+{ }^{9} C_{9}=2^{9}-1}.

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Pankaj

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