In how many ways can 3 different trophies be awarded to 10 students?

In how many ways can 3 different trophies be awarded to 10 students?

Correct Answer: 720

Step 1: Understand that we have 3 different trophies to give out.
Step 2: Recognize that there are 10 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 since we are choosing 3 students from 10 and the order matters. The formula is nPr = n! / (n - r)!, where n is the total number of students and r is the number of trophies.
Step 5: Plug in the numbers: n = 10 (students) and r = 3 (trophies). So we calculate 10P3 = 10! / (10 - 3)!.
Step 6: Calculate 10! = 10 × 9 × 8 × 7! and (10 - 3)! = 7!. So we can simplify: 10P3 = (10 × 9 × 8 × 7!) / 7!.
Step 7: The 7! cancels out, leaving us with 10 × 9 × 8.
Step 8: Calculate 10 × 9 = 90, then 90 × 8 = 720.
Step 9: Conclude that there are 720 different ways to award the 3 trophies to the 10 students.

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