Posted by Anonymous on .
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?

Algebra 
MathMate,
8 choose 3 = C(8,3)=8!/(3!5!)

Algebra 
Anonymous,
What do you mean by C(8,3) ? What does the exclamation point mean?

Algebra 
MathMate,
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 overcount by 6 times, so we divide by 3! to get 8!/(5!3!). 
Algebra 
Anonymous,
Thank you for the explanations!
I also have another question. Where did the 5! came from? 
Algebra 
MathMate,
5! comes from 83.
The general formula for C(n,r) is
C(n,r)=n!/(r!(nr)!)
So if n=8, r=3, nr=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!).