how many ways can we assign the professors

A university needs to assign professors to 6 math courses. there are 3 professors available and all of the courses are taught at different times. In how many ways can the professors be assigned to each professor.

A) there are 6 different courses and there are no restrictions on how many courses can be assigned to each professor

should this just be 6!

B) there are 6 different courses and each professor must be assigned exactly 2 classes.

consider the 2 courses as a repeat, we have 6!/(2!2!)

need someone to verify this for me

Re: how many ways can we assign the professors

Quote:

Originally Posted by

**wopashui** A university needs to assign professors to 6 math courses. there are 3 professors available and all of the courses are taught at different times. In how many ways can the professors be assigned to each professor.

A) there are 6 different courses and there are no restrictions on how many courses can be assigned to each professor

B) there are 6 different courses and each professor must be assigned exactly 2 classes.

This question is really vague!

a) If we take the phrase "no restrictions" quite literally the answer is $\displaystyle 3^6$. That is the number of functions from a set of six to a set of three. But of course that means one professor **might** be assigned all six.

One the other hand, if each prof. must have at least one course then the answer is 540. That is the number of surjections from a set of six to a set of three.

B) This a much easier question.

The answer is $\displaystyle \frac{6!}{2^3}$.

That is the number of ways to arrange the string $\displaystyle AABBCC$.

For example the arrangement, $\displaystyle ABCCAB$ means Prof A teaches the first and fifth courses; Prof B teaches the second and sixth

course; etc.

Re: how many ways can we assign the professors

Hello, wopashui!

Quote:

A university needs to assign 3 professors to 6 different math courses.

All of the courses are taught at different times.

In how many ways can the professors be assigned to each professor:

A) if there are **no restrictions on how many courses**

. . **can be assigned to each professor**.

I see no vagueness.

It states clearly that all six courses could be assigned to one professor.

Each course has a choice of 3 professors.

The number of ways is: .$\displaystyle 3^6 \,=\,729$

Quote:

B) if each professor must be assigned exactly 2 classes.

This is an "ordered partition" problem.

The number of ways is: .$\displaystyle {6\choose2,2,2} \:=\:\frac{6!}{2!\,2!\,2!} \:=\:90$

Re: how many ways can we assign the professors

should the first part be 6^3 instead, sine we are assigning profs to classes, not classes to profs

Re: how many ways can we assign the professors

Quote:

Originally Posted by

**wopashui** should the first part be 6^3 instead, sine we are assigning profs to classes, not classes to profs

How did I know that you were vague on this question?

$\displaystyle 6^3$ is the number of ways to assign each element of a set of three to __one__ element in a set of six.

Doing it that way, each professor is assigned exactly one course.

That means that three courses are left uncovered.

Do you think that is what the question means?

Re: how many ways can we assign the professors

then we can assign the remaining 3 courses again to the three profs, will this work?

Re: how many ways can we assign the professors

Quote:

Originally Posted by

**wopashui** then we can assign the remaining 3 courses again to the three profs, will this work?

Actually, I was trying to show you that your reading does not work. I should have been clearer. Thinking of this as *assigning the professors to the classes* in $\displaystyle 6^3$ ways means two professors can be assigned to the same class. That is clearly not what the question is about.

Without, any restrictions the answer to part A) is $\displaystyle 3^6$.

That is the only way that all six courses have a instructor of record.

Re: how many ways can we assign the professors

Pizza Pizza has 12 different deliveries to make and 3 drivers. In how many ways can the deliveries be made by the 3 drivers so that one driver makes 3 deliveries, another driver makes 5 deliveries, and the third driver makes the remaining deliveries?

this is an example given in class, we have this as 12!/(3!5!4!), so i was thinking we can have 6! for the profs question

Re: how many ways can we assign the professors

Quote:

Originally Posted by

**wopashui** Pizza Pizza has 12 different deliveries to make and 3 drivers. In how many ways can the deliveries be made by the 3 drivers so that one driver makes 3 deliveries, another driver makes 5 deliveries, and the third driver makes the remaining deliveries?

this is an example given in class, we have this as 12!/(3!5!4!), so i was thinking we can have 6! for the profs question

Well yes if each professor teaches two of those courses:

$\displaystyle \frac{6!}{2^3}$