Question: The board of directors of a pharmaceutical corporation has 10 members. An upcoming stockh olders meeting is scheduled to approve a new committee of company officers (chosen from the 10 board members).

a) How many different committees consisting of a president, vice president, secretary, and treasurer can the board present to the stockholders for their approval?

b)Three members of the board are physicians. How many committees from part (a) will have: I. a physician nominated for presidency?

II.exactly one physician appearing on the committee?

III.at least one physician appearing on the committee?

I am not able to understand part b of this question because it is not clarified who are the physicians among the ten people.

Furthermore, if all possible committees in part a have exactly $3$ physicians(no matter how do I chose the persons, since no information is given about them), means that we have $9$ (at least) physicians out of $10$ persons which contradicts the 'exactly $3$' part.

  • $\begingroup$ Also you need to know if one person can have several different posts or not. $\endgroup$ – mathreadler Nov 29 '15 at 14:55
  • $\begingroup$ No they cannot have several different posts $\endgroup$ – Jinn Nov 29 '15 at 14:57
  • $\begingroup$ There is no requirement that the committees in part (a) have exactly $3$ physicians. Indeed, once you get to part (b), where you know that there are exactly $3$ physicians on the whole board, it becomes quite clear that there are many committees that do not have exactly $3$ physicians. Indeed, there are $7\cdot6\cdot5\cdot4$ committees that have no physician. $\endgroup$ – Brian M. Scott Nov 30 '15 at 10:39

The information about who exactly is a physician and who is not isn't important.

Let's take a simplified example. I show you a closed bag and say that inside there are 10 colored balls, numbered from one to ten. 7 of the balls are blue and 3 are red. If I use them to make all possible combinations of 4 balls, how many of them have exactly 3 red balls?

Does your answer change if I first open the bag and show you how the blue and red balls are numbered?

if all possible committees in part a have exactly 3 physicians(no matter how do I chose the persons, since no information is given about them)

As shown above, that is not true. You might be thinking that for whatever combination you can come up with, you can just decide afterwards that all the 3 physicians are in that combination. You can do that, but then there will exist another combination where there are fewer physicians.

An example: Let's say that if the board members are numbered from 1 to 10 and you have a combination 2, 6, 7 and 9. You can then decide that people numbered 2, 6 and 7 are physicians. But then you have many other combinations that don't include those people, for example 1, 3, 4 and 8. Since you've already said that 2, 6 and 7 are physicians, you can't say that anyone in the other group are, because that would bring the total number of physicians to more than 3.

In other words, a person either is or is not a physician. You can't change that status like a light switch or transfer it to another person. Since there is no information given about who exactly the pysicians are, you can assign that status arbitrarily, but you have to do that before you divide the board members into different combinations.

  • $\begingroup$ The assumption that all of the possible committees have $3$ physicians lead to a total of $9$ physicians in all, two specific combinations cannot conclude this. To prove this, assume that we have at most $8$ physicians , then form a committee of $4$ by choosing the $2$ non physicians and any other two, this committee have only two physicians which contradicts the initial assumption. $\endgroup$ – Jinn Nov 29 '15 at 15:04
  • $\begingroup$ Your question specifically says that only 3 people out of the initial pool of 10 are physicians. Having 9 pysicians in all is impossible, because there are only 3. $\endgroup$ – Moyli Nov 29 '15 at 15:07
  • $\begingroup$ That is specifically my question, is this condition of the question consistent with itself? Namely we have exactly $3$ physicians in every committee. $\endgroup$ – Jinn Nov 29 '15 at 15:11
  • $\begingroup$ Sorry, but I don't understand what you're asking. There are not exactly 3 physicians in every possible committee, as I have shown in the answer. I don't understand the logic of how you've come to an opposite conclusion, so I can't offer much additional help. $\endgroup$ – Moyli Nov 29 '15 at 15:14
  • $\begingroup$ Or are you confused about the wording of the question? Note that it says "Three members of the board are physicians". Not members of committees. There is no requirement anywhere that all committees in part a) must have 3 physicians. $\endgroup$ – Moyli Nov 29 '15 at 15:18

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