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 determine the number of different ways the Ds and Rs can be recorded, we can consider it as a combination problem. Since there are 5 cups, Sandra can either record D (for diet pop) or R (for regular pop) for each cup.

For the first cup, there are 2 options (D or R).
Similarly, for the second cup, there are 2 options (D or R).
And so on, for all 5 cups.

Since Sandra records her guesses independently for each cup, we can multiply the number of options for each cup together to find the total number of different ways the Ds and Rs can be recorded:

Total number of different ways = (Number of options for cup 1) * (Number of options for cup 2) * ... * (Number of options for cup 5)

Therefore, the total number of different ways can be calculated as 2 * 2 * 2 * 2 * 2 = 2^5 = 32. Hence, 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, this changes the calculation slightly. We can use a similar approach but with some restrictions.

For the first 3 cups, Sandra can only choose D (for diet pop) since there are exactly 3 diet pops.
For the remaining 2 cups, Sandra can only choose R (for regular pop) since there are exactly 2 regular pops.

So, the possibilities for the first 3 cups would be DDD and for the remaining 2 cups would be RR.

Therefore, there is only 1 way to record the Ds and Rs in this case.