In how many ways can 4 different prizes be distributed among 10 students? (2023)

In how many ways can 4 different prizes be distributed among 10 students? (2023)

Step 1: Understand that we have 4 different prizes to give away.
Step 2: Recognize that there are 10 students who can receive these prizes.
Step 3: Realize that the order in which we give the prizes matters because they are different.
Step 4: Use the formula for permutations since we are choosing 4 students from 10 to receive the prizes. The formula is nPr = n! / (n - r)!, where n is the total number of students and r is the number of prizes.
Step 5: Plug in the numbers: n = 10 (students) and r = 4 (prizes). So we calculate 10P4 = 10! / (10 - 4)! = 10! / 6!.
Step 6: Simplify the calculation: 10! = 10 × 9 × 8 × 7 × 6! and 6! cancels out, leaving us with 10 × 9 × 8 × 7.
Step 7: Calculate the product: 10 × 9 = 90, then 90 × 8 = 720, and finally 720 × 7 = 5040.
Step 8: Conclude that there are 5040 different ways to distribute the 4 prizes among the 10 students.

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