Expert: Sherry Wallin - 12/13/2010

Question
QUESTION: Robert claims that he can always tell the difference between diet pop and regular pop.To test his claim,his friend,Teddy,set up an experiment.Teddy lines up 5 cups of pop(labelled Cup1,Cup2,Cup3,Cup4,Cup5) on a table.Robert then tastes each cup and records D if he thinks it is diet pop or R if he thinks it is a regular pop on a sheet of paper.
___How many different ways can the Ds and Rs be recorded,if Robert guesses which cups have diet pop and which cups have a regular pop?

ANSWER: Andrea~
2^5 = 32 ways because in the first cup it is either D or R so there are 2 things chosen one at a time, i.e. 2C1 and the same goes for each of the other cups so there are 2C1^5 = 2^5 = 32 ways.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: If Robert knew there were 3 cups of diet pop and 2 cups of regular popon the table,how many different ways can the Ds and Rs be recorded?
         Thanks!

Answer
Andrea~

It doesn't matter when in the sequence a certain order is found so think of it this way. There are 3 ways to choose 1 diet pop 3C1 and two ways to choose 1 regular pop 2C1 so there are 5 choices for the first position 5C1, then there are only 2 ways to choose a diet pop, 2C1 and 2 ways to choose a regular pop, 2C1, or 3 ways to choose a diet pop 3C1 and 1 way to choose a regular pop, 1C1. In the next phase there is either 3 ways to choose a diet pop, 3C1 and no way to choose a regular or there is 2 ways to choose a diet pop, 2C1 and only one way to choose a regular pop. Either way there is only one more choice left for a grand total of 10 choices.
I've made another diagram so you can easily see what happens.

Math Prof