In how many ways can 4 different prizes be distributed among 3 students?

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Question: In how many ways can 4 different prizes be distributed among 3 students?

Options:

Correct Answer: 81

Solution:

Each prize can go to any of the 3 students, so the total ways = 3^4 = 81.

In how many ways can 4 different prizes be distributed among 3 students?

In how many ways can 4 different prizes be distributed among 3 students?

Step 1: Identify the number of prizes. In this case, there are 4 different prizes.
Step 2: Identify the number of students. Here, there are 3 students.
Step 3: Understand that each prize can be given to any of the 3 students.
Step 4: For the first prize, there are 3 choices (Student 1, Student 2, or Student 3).
Step 5: For the second prize, there are also 3 choices (Student 1, Student 2, or Student 3).
Step 6: Repeat this for the third prize, which again has 3 choices.
Step 7: Do the same for the fourth prize, which also has 3 choices.
Step 8: Since the choices for each prize are independent, multiply the number of choices together: 3 (for the first prize) * 3 (for the second prize) * 3 (for the third prize) * 3 (for the fourth prize).
Step 9: This can be written as 3^4 (3 raised to the power of 4).
Step 10: Calculate 3^4, which equals 81. Therefore, there are 81 different ways to distribute the 4 prizes among the 3 students.

Combinatorial Distribution – The problem involves distributing distinct items (prizes) to distinct recipients (students), where each item can go to any recipient.