In how many ways can 4 different prizes be awarded to 10 students?

In how many ways can 4 different prizes be awarded to 10 students?

Correct Answer: 5040

Step 1: Understand that we have 4 different prizes to give out.
Step 2: Recognize that there are 10 students who can receive these prizes.
Step 3: Realize that the order in which we award the prizes 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 items (students) and r is the number of items to choose (prizes).
Step 5: In this case, n = 10 (students) and r = 4 (prizes).
Step 6: Plug the values into the formula: P(10, 4) = 10! / (10 - 4)!.
Step 7: Calculate (10 - 4) = 6, so we have P(10, 4) = 10! / 6!.
Step 8: Calculate 10! (10 factorial) which is 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
Step 9: Calculate 6! (6 factorial) which is 6 × 5 × 4 × 3 × 2 × 1.
Step 10: Simplify the expression: P(10, 4) = (10 × 9 × 8 × 7 × 6!) / 6!.
Step 11: The 6! cancels out, leaving us with 10 × 9 × 8 × 7.
Step 12: Calculate 10 × 9 = 90, then 90 × 8 = 720, and finally 720 × 7 = 5040.
Step 13: Conclude that there are 5040 different ways to award the 4 prizes to the 10 students.

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