Question

How many ways can a teacher give five different prizes to five of her 24 students?

Answer 1

There are 24,024 ways a teacher can give five different prizes to five of her 24 students.

Step-by-step explanation:

To find the number of ways a teacher can give five different prizes to five of her 24 students, we can think of it as a permutation problem. Since the order in which the prizes are given matters (each prize is unique), we can use the formula for permutations. The formula for permutations is nPk = n! / (n-k)!, where n is the total number of items and k is the number of items chosen. In this case, n = 24 and k = 5, so the number of ways is 24P5 = 24! / (24-5)!.

We can simplify the expression by calculating the factorial of 24 and 19. Then, we can divide the factorial of 24 by the factorial of 19 to get the final answer. Let's do the calculations:

24P5 = (24 imes 23 imes 22 imes 21 imes 20 imes 19!) / 19!

Cancelling out the 19! on both sides, we get:

24P5 = 24 imes 23 imes 22 imes 21 imes 20

Calculating the product, we find that there are 24,024 ways the teacher can give five different prizes to five of her 24 students.

Answer 2

There are 5100480 ways for a teacher to give five different prizes to five of her 24 students.

This is calculated using the combination formula, which is the number of ways to choose a group of r objects out of a set of n objects, without regard to order. In this case, the teacher has 24 students (n) and wants to choose 5 of them (r) to give prizes to.

The formula for combinations is:

nCr = n! / (r! * (n-r)!)

where:

n is the total number of objects

r is the number of objects to choose

! is the factorial symbol, which means the product of all positive integers from 1 to n

Plugging in the values for this problem, we get:

24C5 = 24! / (5! * (24-5)!)

= 24! / (5! * 19!)

= 24 x 23 x 22 x 21 x 20 / (5 x 4 x 3 x 2 x 1)

= 5100480

Therefore, there are 5100480 ways for the teacher to distribute the prizes.