Two distinct numbers a and b are chosen randomly from the set {2, 2², 2³, 2⁴, ------ 2²⁵} (25 numbers are there). Find the probability that log_a^b (log b to the base a) is an integer.

For Log_a^b to be integer, b>a. So, obviously b cannot be 2 because a and b are distinct. So, if b=2, a should be greater than that, and then it will not be an integer. So if a=2 ,then possible values of b are {2²,2³,2⁴,------2²⁴,2²⁵} There are 24 possible values. Similarly, if a=2², then possible values of b are {2⁴,2⁶,2⁸,------2²²,2²⁴}, i.e. there are 11 possible values. Similarly, if a=2³, then possible values of b are {2⁶,2⁹,2¹²,------2²¹,2²⁴}, i.e. there are 7 possible values. Similarly, if a=2⁴, then possible values of b are {2⁸,2¹²,2¹⁶,------2²⁰,2²⁴}, i.e. there are 5 possible values. Similarly, if a=2⁵, then possible values of b are {2¹⁰,2¹⁵,2²⁰,------2²⁵}, i.e. there are 4 possible values. Similarly, if a=2⁶,then possible values of b are {2¹²,2¹⁸,2²⁴}, i.e. there are 3 possible values. Similalry, if a=2⁷, then possible values of b are {2¹⁴,2²¹}, i.e. there are 2 possible values. Similalry, if a=2⁸, then possible values of b are {2¹⁶,2²⁴}, i.e. there are 2 possible values. Similarly, if a=2⁹, then possible values of b are {2¹⁸), i.e. only 1 possible value. For a=2¹⁰,2¹¹,2¹², the possible values are 2²⁰,2²²,2²⁴ respectively. After that, there is no case possible. So, total possibilities are (24+11+7+5+4+3+2+2+1+1+1+1)= 62 cases Total number of possibilities = a can be 25 numbers and b can be 24 numbers (As a and b should be distinct) Total possibilities are 25*24. Probability = 62/(25*24) = 31/300 (Correct Answer)