Question

In how many ways can 4 prizes be distributed among 5 students, when

• no student gets more than one prize?
  
• a student may get any number of prizes?
  
• no student gets all the prizes?
  

Answer 1

Part $\left(1\right)$:- In this part the first prize can be given to nay $5$ students. But the second prize can be given to only any the left $4$ prize to the left $3$.

Then the number of way of distribution is $=5!=5\times 4\times 3\times 2\times 1=120$ ans.

Part$\left(2\right)$:- In this part first prize can be two any $5,$ second may also be given to any $5$ and so on.

Then the number of ways of distribution here is $=5\times 5\times 5\times 5=625$ ans.

Part$\left(3\right)$:- Here two prizes can be given to the same student but not the fourth one number of ways.$=5\times 5\times 5\times 4=500$ ans.