find out the number of ways in which 5 prizes can

Question

Find out the number of ways in which 5 prizes can be distributed among 5 students such that 

(a)   Each student may get a prize.

(b)  There is no restriction to the number of prizes a student gets.  (8)  

 Ans : (a)  Number of ways in which 5 prizes can be distributed between 5 students in such a way that each one may acquire a prize is 5*4*3*2*1 = 5! ways.

(b) While no restriction on the number of prizes a student may get, after that a student may get either 0 or 1 or 2 or...upto 5 prizes. It depicts that a student have six possible ways in which it may get a prize. Hence the number of ways in which prizes can be distributed is 6⁵.