How many ways can 5 different prizes be awarded to 3 students if each student ca

How many ways can 5 different prizes be awarded to 3 students if each student can receive more than one prize?

How many ways can 5 different prizes be awarded to 3 students if each student can receive more than one prize?

Correct Answer: 243

Step 1: Understand that we have 5 different prizes to give out.
Step 2: Recognize that there are 3 students who can receive these prizes.
Step 3: Note 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 all 5 prizes. Each prize has 3 choices.
Step 7: Since the choices for each prize are independent, multiply the number of choices together: 3 choices for Prize 1, 3 choices for Prize 2, and so on.
Step 8: This means you calculate 3 * 3 * 3 * 3 * 3, which is the same as 3 raised to the power of 5 (3^5).
Step 9: Calculate 3^5, which equals 243.
Step 10: Conclude that there are 243 different ways to award the 5 prizes to the 3 students.

Combinatorics – The problem involves counting the number of ways to distribute distinct prizes among students, which is a combinatorial problem.
Exponential Counting – Each prize can be awarded to any of the 3 students independently, leading to the use of exponentiation (3^5) to find the total combinations.