In this task, you need to provide the correct option for a given problem from the provided options.

Input: Consider Input: Problem: the marks obtained by vijay and amith are in the ratio 4 : 7 and those obtained by amith and abhishek in the ratio of 3 : 2 . the marks obtained by vijay and abhishek are in the ratio of ?
Options: a ) 6 : 8 , b ) 6 : 1 , c ) 6 : 5 , d ) 6 : 7 , e ) 6 : 3

Output: d

Input: Consider Input: Problem: a certain university will select 1 of 4 candidates eligible to fill a position in the mathematics department and 2 of 10 candidates eligible to fill 2 identical positions in the computer science department . if none of the candidates is eligible for a position in both departments , how many different sets of 3 candidates are there to fill the 3 positions ?
Options: a ) 42 , b ) 70 , c ) 140 , d ) 180 , e ) 315

Output: d

Input: Consider Input: Problem: what is the greatest positive integer n such that 3 ^ n is a factor of 54 ^ 100 ?
Options: a ) 100 , b ) 200 , c ) 300 , d ) 600 , e ) 900
Output: c