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?
To solve this problem, we can use the concept of combinations.
a. In this case, Sandra can record either "D" or "R" for each of the 5 cups independently. Therefore, the number of different ways Sandra can record the Ds and Rs is simply 2^5 = 32. This is because for each cup, Sandra has 2 choices (D or R), and she makes an independent choice for each cup.
b. In this case, we know that there are 3 cups of diet pop and 2 cups of regular pop. We need to find the number of ways to arrange these 5 cups with this information.
To calculate this, we can use the concept of combinations. The formula for calculating combinations is nCr, where n is the total number of items and r is the number of items we want to select.
In this case, we have 5 cups in total, and we want to select 3 cups of diet pop (or 2 cups of regular pop, since the remaining 2 will be regular). Therefore, the number of ways to arrange them is:
5C3 * 2C2 = (5! / (3!(5-3)!)) * (2! / (2!(2-2)!)) = (5! / (3!2!)) * (2! / (2!0!)) = (5 * 4 * 3) / (3 * 2 * 1) * (1) = 10 * 1 = 10.
So, there are 10 different ways Sandra can record the Ds and Rs given that there are 3 cups of diet pop and 2 cups of regular pop on the table.