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

Algebra ->  Probability-and-statistics -> 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      Log On


   

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) About Me  (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