How many ways can 4 prizes be distributed among 10 students if each student can

How many ways can 4 prizes be distributed among 10 students if each student can receive at most one prize? (2019)

How many ways can 4 prizes be distributed among 10 students if each student can receive at most one prize? (2019)

Step 1: Understand that we have 4 prizes to give away.
Step 2: Know that there are 10 students who can receive these prizes.
Step 3: Realize that each student can only receive one prize.
Step 4: Since the order in which we give out the prizes matters (first prize, second prize, etc.), we will use permutations.
Step 5: The formula for permutations 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 6: In this case, n = 10 (students) and r = 4 (prizes).
Step 7: Plug the numbers into the formula: 10P4 = 10! / (10 - 4)! = 10! / 6!.
Step 8: Calculate 10! = 10 × 9 × 8 × 7 × 6! and notice that the 6! cancels out.
Step 9: This leaves us with 10 × 9 × 8 × 7.
Step 10: Calculate the result: 10 × 9 = 90, then 90 × 8 = 720, and finally 720 × 7 = 5040.
Step 11: Conclude that there are 5040 ways to distribute the 4 prizes among the 10 students.

Permutations – The problem involves calculating the number of ways to arrange a subset of items (prizes) from a larger set (students) where order matters and no item can be repeated.
Combinatorial Counting – Understanding how to count distinct arrangements or selections from a finite set, particularly when constraints (like maximum prizes per student) are applied.