document.write( "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?\r
\n" ); document.write( "\n" ); document.write( "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.\r
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Algebra.Com's Answer #389240 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Suppose these are the grades on 250 test answer sheets.  Suppose the professor\r\n" );
document.write( "has these 250 test answer sheets in a stack arranged from best test paper to\r\n" );
document.write( "worst.  Now suppose the professor has 4 sheets of paper to use as divider\r\n" );
document.write( "sheets between grades and writes on them:\r\n" );
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document.write( "Divider sheet #1. This sheet is placed between the lowest A and the highest B.\r\n" );
document.write( "Divider sheet #2. This sheet is placed between the lowest B and the highest C.\r\n" );
document.write( "Divider sheet #3. This sheet is placed between the lowest C and the highest D.\r\n" );
document.write( "Divider sheet #4. This sheet is placed between the lowest D and the highest F.  \r\n" );
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document.write( "Now it may be that no one makes a certain grade.  If, say, there are no C's,\r\n" );
document.write( "but there are B's and D's, he will place divider sheets #2 and #3 together\r\n" );
document.write( "between the lowest B and the highest D test papers. Suppose no one makes an A,\r\n" );
document.write( "then he will place divider sheet #1 on the top of the stack. If everybody makes\r\n" );
document.write( "F he will place all 4 divider sheets in order on the top of the stack.  If\r\n" );
document.write( "everybody makes A he will place all 4 sheets on the bottom of the stack in\r\n" );
document.write( "order.  If everybody either makes A or F, then all 4 divider sheets will be in\r\n" );
document.write( "order between the lowest A and the highest F. Hopefully, you get the idea.\r\n" );
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document.write( "The reasoning is:\r\n" );
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document.write( "After placing the 4 divider sheets, the professor now has a stack of 254 sheets\r\n" );
document.write( "of paper, 250 tests and 4 divider sheets.\r\n" );
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document.write( "There are 254 sheets of paper, and the 4 divider sheets can possibly appear in\r\n" );
document.write( "any of the 254 positions, so that's why the answer is C(254,4).\r\n" );
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document.write( "Edwin

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