A lab is developing a new medication for headaches. To do so,they must choose from 10 active ingredients, 5 preservatives and 2 presentations (capsules or tablets). A tablet or a capsule must have at least one active ingredient and one preservative.
a)How many different medications with one active ingredient and two preser-
vatives can the lab produce?
b)How many different medications with two active ingredients and one
preservative can the lab produce?
c)How many different medications with one active ingredient and at most
two preservatives can the lab produce?
For a) there are 10 choices for the active ingred., and 5C2 (5 choose 2) ways to select the preservative, times the number of presentations. Now use the multiplication rule.
For b) there are 10C2 ingred., and 5C2 preserv. times the number of presentations.
For c) there are 10C1 ways to select the ingredient, 5C1 + 5C2 ways for the preserv, and 2 presentations.
Note: nCm means n choose m.
To solve these questions, we need to use combinations and the multiplication rule.
a) To calculate the number of different medications with one active ingredient and two preservatives, we have 10 choices for the active ingredient and we need to select 2 preservatives from a set of 5. This can be calculated using combinations as 5C2. Additionally, there are 2 presentations (capsules or tablets). So, the total number of different medications can be found using the multiplication rule:
Number of medications = 10 * (5C2) * 2
b) For medications with two active ingredients and one preservative, we have 10 choices for the first active ingredient, 9 choices for the second active ingredient (since we need two different ones), and 5 choices for the preservative. Again, there are 2 presentations. So, we can calculate the number of different medications as follows:
Number of medications = (10 * 9) * 5 * 2
c) To find the number of different medications with one active ingredient and at most two preservatives, we have 10 choices for the active ingredient. For the preservative, we can have either one or two preservatives. There are 5C1 ways to select one preservative and 5C2 ways to select two preservatives. Again, there are 2 presentations. So, we can calculate the number of different medications as:
Number of medications = 10 * (5C1 + 5C2) * 2
Note: nCm means selecting m items from a set of n items.