document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151727>Question 1151727</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bag contains four red marbles, one green one, one lavender one, three yellows,<BR>\n" );
document.write( "and two orange marbles. <BR>\n" );
document.write( "How many sets of five marbles include at most one of the yellow ones? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773608 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=773608\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If there is a way to distinguish between the marbles of the same color, say there's a numeric label on each marble, then the answer is 756 as Edwin has shown. \r<BR>\n" );
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document.write( "If there isn't a way to tell two red marbles apart, and similar for the other colors, then the answer is 23.\r<BR>\n" );
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document.write( "Below shows the work on how to get 23.\r<BR>\n" );
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document.write( "==================================================================\r<BR>\n" );
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document.write( "We have this multiset<BR>\n" );
document.write( "{R, R, R, R, G, L, Y, Y, Y, A, A}<BR>\n" );
document.write( "where<BR>\n" );
document.write( "R = red marble<BR>\n" );
document.write( "G = green marble<BR>\n" );
document.write( "L = lavender marble<BR>\n" );
document.write( "Y = yellow marble<BR>\n" );
document.write( "A = orange marble (I'm using the letter \"A\" instead of the letter \"O\" because it is easilly mixed up with the number zero)\r<BR>\n" );
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document.write( "We can write this shorthand notation<BR>\n" );
document.write( "4R, 1G, 1L, 3Y, 2A<BR>\n" );
document.write( "to indicate 4 red, 1 green, 1 lavender, 3 yellow, 2 orange\r<BR>\n" );
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document.write( "We want to select five marbles such that we only have at most 1 yellow marble. This means we either have 1 yellow or no yellows. So we can toss out 2 of the yellow marbles getting this updated notation<BR>\n" );
document.write( "4R, 1G, 1L, 1Y, 2A <BR>\n" );
document.write( "We are able to toss any two yellow marbles because we can't tell them apart.\r<BR>\n" );
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document.write( "Now consider the number of ways we can add to 5 using positive integers. There are 6 ways to do this<BR>\n" );
document.write( "4+1<BR>\n" );
document.write( "3+2<BR>\n" );
document.write( "3+1+1<BR>\n" );
document.write( "2+2+1<BR>\n" );
document.write( "2+1+1+1<BR>\n" );
document.write( "1+1+1+1+1<BR>\n" );
document.write( "These are considered partitions and will help us figure out how to select the marbles in an organized way.\r<BR>\n" );
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document.write( "Partition 4+1<BR>\n" );
document.write( "We have 4 red marbles. No other color has this many marbles. So the '4' applies to red only. The '1' applies to any of the other colors. We have 4 other colors, so we have 4 ways to fill out this partition<BR>\n" );
document.write( "4R,1G<BR>\n" );
document.write( "4R,1L<BR>\n" );
document.write( "4R,1Y<BR>\n" );
document.write( "4R,1A\r<BR>\n" );
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document.write( "Partition 3+2<BR>\n" );
document.write( "No color has 3 marbles other than red, so the '3' applies to the red marbles. No other color than orange has 2 marbles (recall we tossed out 2 yellows leaving with just 1), so the '2' applies to orange<BR>\n" );
document.write( "There is only one way we can have 3 marbles of one color and 2 marbles of another color<BR>\n" );
document.write( "3R, 2A\r<BR>\n" );
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document.write( "Partition 3+1+1<BR>\n" );
document.write( "The '3' applies to the red marbles. The other '1's are for the other colors<BR>\n" );
document.write( "3R,1Y,1A<BR>\n" );
document.write( "3R,1L,1A<BR>\n" );
document.write( "3R,1L,1Y<BR>\n" );
document.write( "3R,1G,1A<BR>\n" );
document.write( "3R,1G,1Y<BR>\n" );
document.write( "3R,1G,1L<BR>\n" );
document.write( "There are 6 ways to fill this section out.\r<BR>\n" );
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document.write( "Partition 2+2+1<BR>\n" );
document.write( "The '2's apply to the red and orange marbles. The '1' is for the other colors<BR>\n" );
document.write( "2R,1Y,2A<BR>\n" );
document.write( "2R,1L,2A<BR>\n" );
document.write( "2R,1G,2A<BR>\n" );
document.write( "There are 3 ways to do it here.\r<BR>\n" );
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document.write( "Partition 2+1+1+1<BR>\n" );
document.write( "The '2' either applies to red marbles or orange marbles. The '1's are for anything else<BR>\n" );
document.write( "2R,1L,1Y,1A<BR>\n" );
document.write( "2R,1G,1Y,1A<BR>\n" );
document.write( "2R,1G,1L,1A<BR>\n" );
document.write( "2R,1G,1L,1Y<BR>\n" );
document.write( "1G,1L,1Y,2A<BR>\n" );
document.write( "1R,1L,1Y,2A<BR>\n" );
document.write( "1R,1G,1Y,2A<BR>\n" );
document.write( "1R,1G,1L,2A<BR>\n" );
document.write( "There are 8 rows in the list above.\r<BR>\n" );
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document.write( "Partition 1+1+1+1+1<BR>\n" );
document.write( "There's only one way to do this selection and that is to simply pick one of each color<BR>\n" );
document.write( "1R,1G,1L,1Y,1A\r<BR>\n" );
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document.write( "Add up the subtotals:<BR>\n" );
document.write( "4+1+6+3+8+1 = 23\r<BR>\n" );
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document.write( "Here are all 23 ways to select five marbles in which we cannot tell similar color ones apart (eg: we can't tell two red marbles apart)<BR>\n" );
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document.write( "<tr><TD >Number</TD><TD >Set</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >4R,1G</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >4R,1L</TD></tr>\r\n" );
document.write( "<tr><TD >3</TD><TD >4R,1Y</TD></tr>\r\n" );
document.write( "<tr><TD >4</TD><TD >4R,1A</TD></tr>\r\n" );
document.write( "<tr><TD >5</TD><TD >3R,2A</TD></tr>\r\n" );
document.write( "<tr><TD >6</TD><TD >3R,1Y,1A</TD></tr>\r\n" );
document.write( "<tr><TD >7</TD><TD >3R,1L,1A</TD></tr>\r\n" );
document.write( "<tr><TD >8</TD><TD >3R,1L,1Y</TD></tr>\r\n" );
document.write( "<tr><TD >9</TD><TD >3R,1G,1A</TD></tr>\r\n" );
document.write( "<tr><TD >10</TD><TD >3R,1G,1Y</TD></tr>\r\n" );
document.write( "<tr><TD >11</TD><TD >3R,1G,1L</TD></tr>\r\n" );
document.write( "<tr><TD >12</TD><TD >2R,1Y,2A</TD></tr>\r\n" );
document.write( "<tr><TD >13</TD><TD >2R,1L,2A</TD></tr>\r\n" );
document.write( "<tr><TD >14</TD><TD >2R,1G,2A</TD></tr>\r\n" );
document.write( "<tr><TD >15</TD><TD >2R,1L,1Y,1A</TD></tr>\r\n" );
document.write( "<tr><TD >16</TD><TD >2R,1G,1Y,1A</TD></tr>\r\n" );
document.write( "<tr><TD >17</TD><TD >2R,1G,1L,1A</TD></tr>\r\n" );
document.write( "<tr><TD >18</TD><TD >2R,1G,1L,1Y</TD></tr>\r\n" );
document.write( "<tr><TD >19</TD><TD >1G,1L,1Y,2A</TD></tr>\r\n" );
document.write( "<tr><TD >20</TD><TD >1R,1L,1Y,2A</TD></tr>\r\n" );
document.write( "<tr><TD >21</TD><TD >1R,1G,1Y,2A</TD></tr>\r\n" );
document.write( "<tr><TD >22</TD><TD >1R,1G,1L,2A</TD></tr>\r\n" );
document.write( "<tr><TD >23</TD><TD >1R,1G,1L,1Y,1A</TD></tr>\r\n" );
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document.write( "The order of the selection does not matter. <BR>\n" );
document.write( "So 1R,1G,1L,1Y,1A is the same as 1G,1R,1L,1Y,1A\r<BR>\n" );
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document.write( "</font> </SPAN>\n" );
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