document.write( "Question 1200865: how many ways can three awards be given to 6 students. each student may receive more than one award? each student may receive no more than one award? \n" ); document.write( "

Algebra.Com's Answer #835086 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "In how many ways can three awards be given to 6 students if
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\n" ); document.write( "\n" ); document.write( " From the context, the students are distinguishable, \r
\n" ); document.write( "\n" ); document.write( " but the awards are indistinguishable.\r
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document.write( "(b)  It is more easy to start from the second question.\r\n" );
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document.write( "     It is the same as to ask, in how many different ways 3 students can be selected from 6 students ?\r\n" );
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document.write( "     It is a classic problem on COMBINATIONS, and the ANSWER is\r\n" );
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document.write( "             in   =  = 5*4 = 20 different ways.\r\n" );
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document.write( "(a)  It is the same as to ask how many solutions this equation\r\n" );
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document.write( "          +  +  +  +  +  = 3   (1)\r\n" );
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document.write( "     has in integer non-negative numbers.\r\n" );
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document.write( "     It is a classic problem to solve using the \"stars and bars\" method.\r\n" );
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document.write( "     The answer is: there are   =  =  = 8*7 = 56\r\n" );
document.write( "     different solutions for equation (1) in non-negative integer numbers.\r\n" );
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document.write( "     So, there are 56 different ways to award 6 students by 3 awards, if each student may receive more than one award \r\n" );
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document.write( "     (the awards are considered as undistinguishable; the students are distinguishable).\r\n" );
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\n" ); document.write( "\n" ); document.write( "For the stars and bars method see this Wikipedia article\r
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\n" ); document.write( "\n" ); document.write( "or read my lesson\r
\n" ); document.write( "\n" ); document.write( " - Stars and bars method for Combinatorics problems \r
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