Andrea, Kate, Verona, Beverely, Karen, Joanna, George, and Paula are starters on their school basketball team. How many different groups of three can be chosen for a newspaper photo?
8 choose 3 = C(8,3)=8!/(3!5!)
What do you mean by C(8,3) ? What does the exclamation point mean?
C(8,3) is the function that represents the number of ways to choose 3 items out of 8. C is short for combination.
8! is a math function call 8 factorial, and stands for the product 8*7*6*5*4*3*2*1=40320
3!=3*2*1, etc.
What C(8,3) means is that to choose 3 people, we would have a choice of 8 candidates for the first one, 7 candidates for the second, and 6 candidates for the third.
However, to choose the same 3 candidates, we could choose them in 6 different orders, namely ABC, ACB, BAC, BCA, CAB, CBA, or 3! ways.
So 8*7*6=8!/5!, but we have to divide by 3! in order not to over-count by 6 times, so we divide by 3! to get 8!/(5!3!).
Thank you for the explanations!
I also have another question. Where did the 5! came from?
5! comes from 8-3.
The general formula for C(n,r) is
C(n,r)=n!/(r!(n-r)!)
So if n=8, r=3, n-r=5, and
C(8,3)=8!/(3!5!)
You will notice that by symmetry,
C(8,3)=C(8,5) because they both equal 8!/(3!5!).
To determine the number of different groups of three that can be chosen for a newspaper photo, we can use the concept of combinations. A combination is a selection of items in which the order does not matter.
In this case, we need to choose 3 players from a group of 8: Andrea, Kate, Verona, Beverely, Karen, Joanna, George, and Paula.
To find the number of combinations, we can use the formula for combination:
C(n, r) = n! / (r! * (n - r)!)
Where:
- n is the total number of items in the group (8 in this case)
- r is the number of items to be selected (3 in this case)
- ! denotes the factorial function, which means multiplying a number by all the positive integers less than it down to 1
Applying the formula, we can calculate the number of combinations:
C(8, 3) = 8! / (3! * (8 - 3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6 * 5!) / (3 * 2 * 1 * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
Therefore, there are 56 different groups of three players that can be chosen for the newspaper photo.