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

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

Step 1: Understand that we have 10 students and we want to award 3 prizes.
Step 2: Recognize that the order in which we award the prizes matters (1st prize, 2nd prize, 3rd prize).
Step 3: Use the formula for permutations since the order is important. The formula is nPr = n! / (n-r)!, where n is the total number of items (students) and r is the number of items to choose (prizes).
Step 4: In this case, n = 10 (students) and r = 3 (prizes).
Step 5: Plug the values into the formula: 10P3 = 10! / (10-3)!.
Step 6: Calculate (10-3) which is 7, so we have 10! / 7!.
Step 7: Simplify 10! / 7! to 10 × 9 × 8 (because 7! cancels out).
Step 8: Calculate 10 × 9 × 8 = 720.
Step 9: Conclude that there are 720 different ways to award the 3 prizes to the 10 students.

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