miguel line sup 5 cups of pop (labelled Cup 1, 2, 3, 4, 5 on a table. sandra then tastes each cup and records "D" if she thinks its diet pop or "R" if she thinks its regular pop.
a. how many different ways can the Ds and Rs be recorded if sandra guesses which cups have diet and which have regular pop?
b. if sandra knew there were 3 cups of diet pop and 2 cups of regular pop on the table, how many different ways can the Ds and Rs be recorded?
a. To find the number of different ways the Ds and Rs can be recorded, we need to consider the possibilities for each cup. Since there are five cups (Cup 1, 2, 3, 4, 5), each cup can either be labeled with "D" for diet pop or "R" for regular pop.
For each cup, there are two possibilities (D or R), so the total number of different ways the Ds and Rs can be recorded is calculated by multiplying the number of possibilities for each cup. In this case, since there are five cups, the total number of different ways is 2^5 (2 to the power of 5) which is equal to 32.
Therefore, there are 32 different ways the Ds and Rs can be recorded if Sandra guesses which cups have diet and which have regular pop.
b. If Sandra knows that there are 3 cups of diet pop and 2 cups of regular pop on the table, the possibilities for the Ds and Rs are constrained. We can think of this problem as selecting 3 cups out of the 5 cups for diet pop and the remaining 2 cups for regular pop.
To calculate the number of different ways, we will use combinations. The number of combinations of selecting 3 items out of 5 is denoted as "5 choose 3" which is calculated as:
5! / (3!(5-3)!) = 5! / (3!2!) = (5 * 4 * 3!) / (3! * 2 * 1) = (5 * 4) / (2 * 1) = 10.
Therefore, there are 10 different ways the Ds and Rs can be recorded if Sandra knows there are 3 cups of diet pop and 2 cups of regular pop on the table.