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?
please show me how to go about this
To solve this problem, we can use combinations and the concept of permutations.
a. In this case, Sandra can record "D" or "R" for each cup, independently, without any restrictions. Since there are 5 cups, Sandra has 2 options for each cup. Thus, the total number of possibilities is calculated by multiplying the options for each cup.
Total possibilities = 2 * 2 * 2 * 2 * 2 = 32
Therefore, there are 32 different ways that Sandra can record "D" or "R" for each cup.
b. In this case, Sandra already knows that there are 3 cups of diet pop and 2 cups of regular pop on the table. We need to consider the arrangements of D and R within those groups.
To calculate the possible arrangements, we can use combinations and permutations:
First, we need to consider the arrangements of the diet pops. Since there are 3 diet pops, Sandra has to choose 3 cups out of the 5 cups to contain diet pop. This can be done using a combination:
Number of ways to choose 3 cups for diet pop = C(5, 3) = 5! / [(3!)(5-3)!] = 5! / (3!2!) = 10
Next, we need to consider the arrangements of the regular pops. Since there are 2 regular pops, Sandra has to choose 2 cups out of the remaining 2 cups to contain regular pop. Again, we can use a combination:
Number of ways to choose 2 cups for regular pop = C(2, 2) = 2! / [(2!)(2-2)!] = 2! / (2!0!) = 1
Now that we have determined the possible arrangements for the diet and regular pops, we can calculate the total number of possibilities by multiplying the two results:
Total possibilities = Number of ways to choose 3 cups for diet pop * Number of ways to choose 2 cups for regular pop
= 10 * 1
= 10
Therefore, there are 10 different ways that Sandra can record "D" or "R" for each cup, given that there are 3 cups of diet pop and 2 cups of regular pop on the table.