How many ways can 5 different prizes be awarded to 3 students?

How many ways can 5 different prizes be awarded to 3 students?

Correct Answer: 243

Step 1: Understand that we have 5 different prizes to give away.
Step 2: Recognize that there are 3 students who can receive these prizes.
Step 3: For each prize, we can choose any of the 3 students to receive it.
Step 4: Since there are 5 prizes, and each prize can go to any of the 3 students, we multiply the choices for each prize.
Step 5: This means for the first prize, there are 3 choices, for the second prize, there are also 3 choices, and so on for all 5 prizes.
Step 6: Therefore, the total number of ways to award the prizes is 3 (choices for prize 1) multiplied by 3 (choices for prize 2) multiplied by 3 (choices for prize 3) multiplied by 3 (choices for prize 4) multiplied by 3 (choices for prize 5).
Step 7: This can be written as 3^5 (3 raised to the power of 5).
Step 8: Calculate 3^5, which equals 243.
Step 9: Conclude that there are 243 different ways to award the 5 prizes to the 3 students.

Combinatorics – The problem involves distributing distinct items (prizes) among distinct recipients (students), which is a fundamental concept in combinatorial mathematics.
Counting Principles – The solution applies the multiplication principle of counting, where each prize can be awarded to any of the students independently.