Permutations and Combinations

Question

How do you solve this? Please include all steps. Thanks!

 

on the tv show american idol, a young rising star is eliminated each week. the first week, each of the 12 contestants sings a song. how many possible ways could the contestants be ordered 1-12? how many possible ways could six men and six women be ordered to alternate female and male contestants?

 

Also, what word clues can you use to figure out if the problem is using permutations, combinations or the fundamental counting principle? Thanks!

Answer 1

Think of this problem in terms of "slots" that contestants are placed into (first slot, second slot, and so on).

 

When a decision is made to place a contestant into the first slot, there are 12 possible choices.  AFTER this choice has been made, it is then compounded with the choice of the next slot which has 11 people to choose from.  This process goes on and on until the last slot is filled (slot twelve) where there is only 1 choice of a contestant.

 

Based on the the idea of compounding choices, each number of choices for one decision should be multiplied by the number of choices for each following decision.  This means that for 12 contestants there are

12*11*10*...*2*3*1 = 12!   possible ways to arrange the contents in a given order.

 

Now lets assume that the order has to alternate going MFMFMFMFMF....

 

If the selection starts with a Female first, then there are 6 choices.  This means then that the following person would have to be Male and there are 6 possible choices for that slot.  On the third slot, it must be female but there has already been one female selected so there are only 5 possible choices.  This process continues until all 12 slots are filled as shown

 

slot                      1  2  3  4   5   6   7   8   9   10   11   12

------------------------------------------------------------------------------

gender                  F  M  F  M  F   M   F   M   F    M    F    M  

------------------------------------------------------------------------------

# of contestants    6  6   5  5  4   4   3   3   2    2     1    1

 

As you can see this can be rearranged to give 6*5*4*3*2*1 * 6*5*....*2*1

which is the same as

 

6! * 6!

 

Overall, this problem uses both the idea of permutations and the fundamental principle of counting.

Answer 2

I really like David's explanation.  I too like to think of these as "slots" that I'm choosing to put things in.  But just to add a few more comments.
 
Notice that the "pool" (choices) for the slots as you follow from left to right can change.  For instance, alternating women and men.  Another example is where you choose some from one pool and then choose some from another pool, like a license plate that has 3 letters and 3 numbers.  Your first 3 slots would have 26 choices.  (Which could be 26*26*26 or 26*25*24 depending if you can repeat.)  Then the next 3 slots would have 10 choices.  (Again, either 10*10*10 or 10*9*8.)  But even though the "pool" changed, you just keep going in order and everything gets multiplied along that path regardless if the pool changed.
 
As to permutation vs combination...
 
A permutation is something where the order you do it in will make a difference.  A lot of people have a tough time understanding what "order matters" means.  For instance, it matters with the contestants because it has to match a specific slot.  If the person who goes 1st goes 5th instead, that's a different option, so the order they go in matters.  It even says "ordered." :-)
 
Other examples of order mattering are license plates, combination locks, putting people into positions (such as president, vice president, etc.).  The combination 3, 7, 2 is not the same as the combination 7, 2, 3.  Person A as pres and person B as VP, is not the same as person B being pres and person A being VP.  This is what is meant by "order matters."
 
A combination is where order does not matter.  Common examples of these are like picking a 5-card hand (it doesn't matter what order you get the cards in) or making a committee (it doesn't matter what order you pick the people in), stuff like that.  Would it matter if you picked people as A, B, C versus C, A, B?  If there's no positions (like president), then no.  Hence, combination.
 
(Those can be combined into one problem by the way.  Watch out for things where 2 have to be in order but the rest don't matter.)