Finding the number of possible selections from more than enough candidates

Intro

Howdy I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find the number of possible selections from more than enough candidates.  Here's our problem statement: A clinical test on humans of a new drug is normally done in three phases.  Phase I is conducted with a relatively small number of healthy volunteers.  For example, a Phase I test of a specific drug involved only seven subjects.  Assume that we want to treat seven healthy humans with this new drug, and we have eleven suitable volunteers available.  Complete Parts A through C below.

Part A

Part A says, “If the subjects are selected and treated in sequence so that the trial is discontinued if anyone displays adverse effects, how many different sequential arrangements are possible if seven people are selected from the eleven that aren't available?  Choose the correct answer below.”  So here we're trying to find the number of possibilities.  We have eleven people to select from, and we only need to select seven.  So we're selecting from eleven seven at a time.

This is a problem involving permutations and combinations.  To help us calculate the permutations and combinations, I have here a calculator from the online website calculator.net.  I'm not getting any kickback money or advertising revenue from calculator.net.  This is totally a freebie.  I'm trying to use something online that I can use in my video.  Normally when I calculate permutations and combinations, I use my calculator to calculate it.  Many calculators have combination and permutation functions built into them, so you'll have to learn how to use the particular model that you have because they're all a little different.  But I'm going to use this online calculator to illustrate how this is actually done.

The big question when working these types of problems is “Do we want to calculate a permutation, or do you want to calculate a combination?” The key determinate in answering that question is the answer to this question: Does the order matter?  If the order in which we arrange the elements matters, then we want to calculate permutations.  If it doesn't matter, then we want to calculate combinations.

Let's look back at our problem statement. Notice here that we're saying we're treating people in sequence.  So because there's a sequence here, that means there's an order that we want to treat people in.  So the order is going to matter.  Therefore, we want to calculate permutations.  At this point I go over to my online calculator.  I'm going to put in the total number of the set, which is 11 because there are 11 people that we want to treat.  The total that we're selecting from is 7, so we're taking 11 people selected 7 at a time.  I hit Calculate, and this is the number of permutations that I'm looking for.  So I go over here to my answer field, select that as our answer option.  Excellent!

Part B

Now Part B says, “If 7 subjects are selected from the 11 that are available, and the 7 selected subjects are all treated at the same time, how many different treatment groups are possible?”  Well, if we're treating all the people at the same time, obviously the order doesn't matter.  So here we want combinations.  So here we're going to calculate combinations.  This is the number we want here from the online calculator; we've already calculated it.  So I'm just going to put that number, 330, into my answer field.  Fantastic!

Part C

And finally Part C — “If 7 subjects are randomly selected and treated at the same time, what is the probability of selecting the 7 youngest subjects?”  Well, remember that probability is simply calculated by finding the part and dividing by the whole.  So we're gonna have a fraction here, so I'm gonna put my fraction element here in my answer field.

The part goes on top.  What's the part?  The part is the seven youngest subjects.  Now out of all the possible outcomes that we could have in our sample space, how many of them have the seven youngest subjects?  There's only going to be one of those outcomes in our sample space that has the seven youngest subjects.  So that is the part that we put at the top of our fraction.

In the bottom of our fraction, we put the whole.  The whole is going to be either the permutations or the combinations because we want the total number that's in our sample space.  But which is it?  Is that the permutations or the combinations?  Well, we're treating people here at the same time.  This means that the order is not important, and therefore we want combinations.  So down here in the whole part of my fraction, I'm going to put the number of combinations that I have.  Well done!

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