In how many ways can 4 prizes be distributed among 10 students?

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Question: In how many ways can 4 prizes be distributed among 10 students?

Options:

Correct Answer: 5040

Solution:

The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.

In how many ways can 4 prizes be distributed among 10 students?

In how many ways can 4 prizes be distributed among 10 students?

Step 1: Understand that we have 4 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 (since each prize is distinct).
Step 4: Use the formula for permutations, which is denoted as nPr, 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), so we need to calculate 10P4.
Step 6: The formula for permutations is nPr = n! / (n - r)!. Here, 10P4 = 10! / (10 - 4)! = 10! / 6!.
Step 7: Calculate 10! (which is 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) and 6! (which is 6 x 5 x 4 x 3 x 2 x 1).
Step 8: Simplify the calculation: 10! / 6! = (10 x 9 x 8 x 7) / 1 = 5040.
Step 9: Conclude that there are 5040 different ways to distribute the 4 prizes among the 10 students.

Permutations – The question tests the understanding of permutations, specifically how to calculate the number of ways to arrange a subset of items (prizes) from a larger set (students).
Combinatorial Counting – It also involves combinatorial counting principles, where the order of selection matters since each prize is distinct.