BlueDefender wrote:
Could somebody please explain how "since the order is not important" translates into "we divide the total by 3! ways"?
If you take three things, say the letters A, B and C, there are 3! = 6 different orders you can put them in:
ABC
ACB
BAC
BCA
CAB
CBA
Now say A, B and C are three of the ten employees of a company (short for Ali, Beka and Carlos, say). You might be asked to count how many ways you can pick three of these ten employees where order is important, or where order is not important. So you might be asked:
• how many selections are possible of a President, Vice-President and Treasurer (if different people occupy each role)? Then order is important, and we have 10 choices for President, 9 for Vice-President, and 8 for Treasurer, for 10*9*8 possibilities in total. Notice when we happen to pick Ali, Beka and Carlos for the three positions, we have 6 different possibilities -- the six I listed above. For example, if we pick them in the order ABC, then Ali is President, Beka is VP, and Carlos is Treasurer, while if we pick them in the order CAB, Carlos is President, Ali is VP, and Beka is Treasurer.
• how many selections are possible for a board of 3 directors? Then order is not important -- we have the same Board no matter what order we list our names in. Now notice that when we happen to pick Ali, Beka and Carlos, there is only one Board we can make. The six possibilities I listed above (ABC, ACB, BAC, BCA, CAB, CBA) are all the same Board. So when order matters, when we pick 3 specific people, we get 3! = 6 different possibilities, but when order does not matter, we only get 1 possibility. When order matters, the number of possibilities is 3! = 6 times larger than when order doesn't matter. So if we counted pretending order matters, we can just divide by 3! to find the number of possibilities when order does not matter.
In general, if you're picking k things, and their order doesn't matter, you can count all your possibilities by pretending first that order does matter, and then dividing by k! to account for the fact that order does not matter.