document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161125>Question 1161125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bag contains 4 red marbles, 3 green ones, 1 lavender one, 3 yellows, and 2 orange marbles.\r<BR>\n" );
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document.write( "How many sets of five marbles include at least three red ones? </SPAN>\n" );
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document.write( "\"At least 3\" means \"3 or more\". Considering there are 4 red marbles, the only possible scenarios are:<BR>\n" );
document.write( "<ul><li>picking exactly 3 red marbles</li><li>picking exactly 4 red marbles</li></ul><BR>\n" );
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document.write( "Let's consider the situation where we have exactly 3 red marbles selected. There are 4 red marbles overall, which means there are 4 ways to form a group of 3 selected red. Put another way, there are 4 ways to not a select a particular red marble. \r<BR>\n" );
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document.write( "Using the combination function, you should find that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _n C _r = 4\"> when n = 4 and r = 3.\r<BR>\n" );
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document.write( "There are 3 green marbles, 1 lavender, 3 yellows and 2 orange giving a total of 3+1+3+2 = 9 marbles that aren't red. Use the combination function to find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _n C _r = 36\"> when n = 9 and r = 2. This says there are 36 ways to select two marbles from a pool of 9.\r<BR>\n" );
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document.write( "Multiply this with the number of ways to pick the red marbles (4) and we get 36*4 = 144\r<BR>\n" );
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document.write( "There are 144 ways to pick exactly 3 red marbles and 2 non-red marbles.\r<BR>\n" );
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document.write( "Let A = 144. We'll use it later.\r<BR>\n" );
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document.write( "We have 4 red marbles. There is only one way to select a group of 4 red marbles if the pool size is also 4. \r<BR>\n" );
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document.write( "There are 9 non-red marbles and only one slot to fill with this pool to pick from. So we have 9 different ways to fill it.\r<BR>\n" );
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document.write( "Overall, there are 1*9 = 9 different ways to pick four red marbles followed by one non-red marble.\r<BR>\n" );
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document.write( "Let B = 9\r<BR>\n" );
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document.write( "The results from the earlier sections are then added up. The values of A and B can be added because the events they represent are mutually exclusive.\r<BR>\n" );
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document.write( "A+B = 144+9 = 153\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>153</font><BR>\n" );
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