Suppose a doctor has the ability to prescribe you 5 types of macrolides and 7 ty

Question

Suppose a doctor has the ability to prescribe you 5 types of macrolides and 7 types of penicillins to combat an infection you have unfortunately come down with.

(A) If the doctor initially recommends you to take 2 of the macrolides sequentially and then take 3 of the penicillins sequentially, how many possible ways could the doctor do this?

(B) The doctor rethinks their plan for your infection. They know are recommending you to take 7 of the drugs (in any combination) all at once (in a cocktail, if you will). How many ways might your doctor be able to do this?

(C) Once again, after another failed treatment, your doctor comes up with one last treatment regime (hopefully, because you're cured). They are now saying to take a cocktail of 3 of the macrolides and then take another cocktail of 2 of the penicillins. How many different ways could your doctor do this?

Explanation / Answer

A. Number of ways = Number of ways of selecting and arranging 2 macrolides from 5 * Number of ways of selecting and arranging 3 penicillins from 7

= (5C2 * 2!) * (7C3 * 3!)

= 4200

B. Number of ways = Number of ways of selecting 7 drugs from 12 = 12C7 = 792

C. Number of ways = Number of ways of selecting 3 macrolides from 5 * Number of ways of selecting 2 penicillins from 7

= 5C3 * 7C2 = 210

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