How many ways can a teacher give six different prizes to six of her 22 students?
Well, let's see here. If the teacher wants to give away six different prizes to six out of her 22 students, it sounds like a perfect opportunity for a "Prize Picking Palooza"!
Now, to calculate the number of ways this can be done, we can use a mathematical concept called combination. We'll use the formula for combination, which is:
C(n, r) = n! / (r!(n-r)!)
Where n represents the total number of possibilities (22 students) and r represents the number of choices (6 prizes).
So, plugging in the numbers, we have:
C(22, 6) = 22! / (6!(22-6)!)
Now, I can't do all the calculations in my funny little head, but I'm sure a calculator can handle it for you. Let's give it a whirl and find out how many ways the prizes can be given!
To find the number of ways a teacher can give six different prizes to six of her 22 students, we will use the concept of combinations.
The number of ways to select six students out of 22 can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
Here, n is the total number of students (22) and r is the number of students to be selected (6).
Let's plug in the values and calculate:
22C6 = 22! / (6!(22-6)!)
= 22! / (6!16!)
To simplify further, let's evaluate the factorials:
22! = 22 x 21 x 20 x 19 x 18 x 17 x 16! (canceling out 16! in the denominator)
22C6 = (22 x 21 x 20 x 19 x 18 x 17) / (6 x 5 x 4 x 3 x 2 x 1)
Calculating this expression, we find:
22C6 = 746,913
So, there are 746,913 different ways for the teacher to give six different prizes to six of her 22 students.
To determine the number of ways a teacher can give six different prizes to six of her 22 students, we can use the concept of permutations.
A permutation represents the arrangement or order of a set of objects. In this case, we want to find the number of permutations of 6 students chosen from a group of 22.
The formula to calculate the permutation is given by:
P(n, r) = n! / (n - r)!
Where n represents the total number of objects in the set (22 students) and r represents the number of objects we want to select (6 prizes).
So, applying the formula, we can calculate the number of ways as follows:
P(22, 6) = 22! / (22 - 6)!
= 22! / 16!
= (22 * 21 * 20 * 19 * 18 * 17 * 16!) / 16!
= (22 * 21 * 20 * 19 * 18 * 17)
= 132,651,600
Therefore, there are 132,651,600 ways the teacher can give six different prizes to six of her 22 students.