How many ways can 3 different trophies be awarded to 5 students? (2022)

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Question: How many ways can 3 different trophies be awarded to 5 students? (2022)

Options:

Correct Answer: 120

Exam Year: 2022

Solution:

The number of ways to award 3 trophies to 5 students is given by P(5,3) = 5! / (5-3)! = 60.

How many ways can 3 different trophies be awarded to 5 students? (2022)

How many ways can 3 different trophies be awarded to 5 students? (2022)

Step 1: Understand that we have 3 different trophies to give out.
Step 2: Recognize that there are 5 students who can receive these trophies.
Step 3: Realize that the order in which we award the trophies matters because they are different.
Step 4: Use the formula for permutations, which is P(n, r) = n! / (n - r)!, where n is the total number of students and r is the number of trophies.
Step 5: In this case, n = 5 (students) and r = 3 (trophies).
Step 6: Plug the numbers into the formula: P(5, 3) = 5! / (5 - 3)!
Step 7: Calculate 5! (which is 5 x 4 x 3 x 2 x 1 = 120) and (5 - 3)! = 2! (which is 2 x 1 = 2).
Step 8: Now divide: 120 / 2 = 60.
Step 9: Conclude that there are 60 different ways to award the 3 trophies to the 5 students.

Permutations – The arrangement of a subset of items from a larger set, where the order matters.
Factorial – The product of all positive integers up to a given number, used in calculating permutations.