Answer :
Final answer:
When no student gets more than one prize, the number of ways to distribute the 4 prizes among 5 students is 70 ways. When a student may get any number of prizes, the number of ways is 625 ways. When no student gets all the prizes, the number of ways is 620 ways.
The correct answer is b) 120, 3125, 5
Explanation:
To solve this problem, we can use the concept of distributing identical items among distinct groups. Let's break down each scenario:
(i) When no student gets more than one prize:
In this case, we are distributing 4 prizes among 5 students such that no student can receive more than one prize. This is a case of distributing identical objects into distinct groups without repetition. The formula to solve this is using "n + r - 1 choose r," where n is the number of objects to distribute and r is the number of groups.
Using the formula, the number of ways to distribute the prizes in this scenario is:
5 + 4 - 1 choose 4 = 8 choose 4 = 70 ways
(ii) When a student may get any number of prizes:
In this scenario, we have the freedom to distribute any number of prizes to each student. This is a case where each prize can be given to any of the 5 students independently.
The number of ways to distribute the prizes in this scenario is:
5 choices for the first prize * 5 choices for the second prize * 5 choices for the third prize * 5 choices for the fourth prize = 5^4 = 625 ways
(iii) When no student gets all the prizes:
For this case, we need to subtract the number of ways where a student gets all the prizes from the total possible ways in scenario (ii).
If a student were to get all the prizes, there is only one way for each student to get all the prizes. Thus, there are 5 ways where a student gets all the prizes.
Therefore, the number of ways where no student gets all the prizes is:
625 total ways - 5 ways = 620 ways
In conclusion, the number of ways for each scenario are:
(i) 70 ways
(ii) 625 ways
(iii) 620 ways
Therefore, the correct options are:
(ii) 120, 625, 4.
