SOLUTION: Your chemistry professor is to assign grades of A, B, C, D, or F in a class of 250 students. How many different grade distributions (total number of A's, B's, C's, D's, and F's; Fo

Question 618813: Your chemistry professor is to assign grades of A, B, C, D, or F in a class of 250 students. How many different grade distributions (total number of A's, B's, C's, D's, and F's; For example, 40 A's, 75 B's, 90 C's, 40 D's, and 5 F's) are possible?
The answer is 169,362,501. I am unsure as to how to arrive to this answer. Looking at a solution to it, they used a combinatorial of picking 4 from a pool of 250. --> C(250+4,4) ....But, it does not explain why.
Please Help, thank you.
Answer by Edwin McCravy(20086) (Show Source): You can put this solution on YOUR website!

Suppose these are the grades on 250 test answer sheets.  Suppose the professor
has these 250 test answer sheets in a stack arranged from best test paper to
worst.  Now suppose the professor has 4 sheets of paper to use as divider
sheets between grades and writes on them:

Divider sheet #1. This sheet is placed between the lowest A and the highest B.
Divider sheet #2. This sheet is placed between the lowest B and the highest C.
Divider sheet #3. This sheet is placed between the lowest C and the highest D.
Divider sheet #4. This sheet is placed between the lowest D and the highest F.  

Now it may be that no one makes a certain grade.  If, say, there are no C's,
but there are B's and D's, he will place divider sheets #2 and #3 together
between the lowest B and the highest D test papers. Suppose no one makes an A,
then he will place divider sheet #1 on the top of the stack. If everybody makes
F he will place all 4 divider sheets in order on the top of the stack.  If
everybody makes A he will place all 4 sheets on the bottom of the stack in
order.  If everybody either makes A or F, then all 4 divider sheets will be in
order between the lowest A and the highest F. Hopefully, you get the idea.

The reasoning is:

After placing the 4 divider sheets, the professor now has a stack of 254 sheets
of paper, 250 tests and 4 divider sheets.

There are 254 sheets of paper, and the 4 divider sheets can possibly appear in
any of the 254 positions, so that's why the answer is C(254,4).

Edwin

RELATED QUESTIONS
Possible grades for a class are A, B, C, D, and F. (No +/− 's.) How many... (answered by macston)

The final grades of students in Foreign Language are normally distributed. records show... (answered by CPhill)

Assume that a grade of A, B, C, D, F, I, or NC is assigned to each of 8 students. In how (answered by macston)

When asked if grades are �curved� Professor Makanyanga replies that she uses the normal... (answered by robertb)

My morning class has 20 students, I want to randomly select 3 students to assign grades... (answered by stanbon)

A professor teaches a class of 60 students and another class of 40 students. Five... (answered by edjones)

Of the students in PHYS 152 at State College last semester, 41 students earned an A, 62... (answered by macston)

I'm having difficulty with these problems if anyone could help I'd really appreciate it. (answered by stanbon)

in a certain college class, each student received a grade of a,b,c,d or incomplete. in... (answered by stanbon)