How many different ways can 4 prizes be distributed among 10 students? (2020)

How many different ways can 4 prizes be distributed among 10 students? (2020)

Step 1: Understand that we have 4 prizes to give away and 10 students to choose from.
Step 2: Recognize that the order in which we give the prizes matters. This means that giving Prize 1 to Student A and Prize 2 to Student B is different from giving Prize 1 to Student B and Prize 2 to Student A.
Step 3: Use the formula for permutations since we are interested in the order of distribution. 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 4: In this case, n = 10 (students) and r = 4 (prizes). So we need to calculate 10P4.
Step 5: Plug the values into the formula: 10P4 = 10! / (10 - 4)! = 10! / 6!.
Step 6: Calculate 10! (10 factorial) which is 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, but since we are dividing by 6!, we can cancel out the 6 × 5 × 4 × 3 × 2 × 1 from both the numerator and denominator.
Step 7: This leaves us with 10 × 9 × 8 × 7 in the numerator.
Step 8: Calculate 10 × 9 = 90, then 90 × 8 = 720, and finally 720 × 7 = 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 different arrangements of the same prizes among students count as different distributions.