First slide

Combinations

Question

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

Moderate

A

$\sum_{i = 1}^{9}^{9} P_{i}$
B

$\sum_{i = 1}^{10}^{9} P_{i}$
C

$2^{9} - 1$
D

$2^{10} - 1$

Solution

A rational number of the desired category is of the form 2015. $x_{1} x_{2} \dots x_{k}$ and $9 \geq x 1 >$ $x 2 > \dots > x k \geq 1 .$ We can choose k digits out 9 in $^{9} C_{k}$
ways and arrange them in de creasing order in just one way. Thus, the desired number of rational numbers is $^{9} C_{1} +^{9} C_{2} + \dots +$ $^{9} C_{9} = 2^{9} - 1$

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