All the five digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude.The $97$th number in the list does not contain the digit
$(A)4\hspace{1 cm}(B)5\hspace{1 cm}(C)7\hspace{1 cm}(D)8$
The question is demanding the numbers of the type of numbers as $''24567''$.Such numbers should start from either 1,2,3,4,5.
But i do not know how to calculate the $97$th number.
Best Answer
Selecting an admissible number means selecting a five element subset of $\{1,2,3,4,5,6,7,8,9\}$.
There are ${8\choose4}=70$ numbers of the form $1x$ and ${7\choose4}=35$ numbers of the form $2x$. Since $70<97<70+35$ the first digit is $2$.
There are ${6\choose 3}=20$ numbers of the form $23x$ and ${5\choose 3}=10$ numbers of the form $24x$. Since $70+20<97<70+20+10$ the first two digits are $24$.
There are ${4\choose2}=6$ numbers of the form $245x$. Since $70+20+6=96$ the number we are after is the first number of the form $246x$, which is $24678$. It follows that (B) is true.