document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202169>Question 1202169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would I solve this problem: “A bag contains 4 red, 6 white and 9 blue marbles. How many ways can 5 marbles be drawn so that *at least* 2 marbles are blue?”\r<BR>\n" );
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document.write( "I have tried the following but wonder if they one is correct:<BR>\n" );
document.write( "1. 9C2 * 17C3   I thought that 2 blue marbles would be drawn, leaving 17 blue, red or white marbles to be drawn. The scenarios would be multiplied to yield 24480 ways.<BR>\n" );
document.write( "2. Sum the results from each scenario of 2-5 blue marbles being drawn. (9C2 * 10C3) + (9C3 * 10C2) + (9C4 * 10C1) + (9C5) = 9486\r<BR>\n" );
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document.write( "Is there a flaw in my thinking? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836823 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=836823\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "9C2*17C3 = 24480 is way too big compared to the upper bound 19C5 = 11628\r<BR>\n" );
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document.write( "11628 is the highest we can go.<BR>\n" );
document.write( "This is one piece of evidence to allow us to rule out 9C2*17C3\r<BR>\n" );
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document.write( "The 2nd calculation you wrote is correct. The answer is <font color=red size=4>9486</font>\r<BR>\n" );
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document.write( "9C2 * 10C3 = number of ways to get 2 blue and 3 non-blue<BR>\n" );
document.write( "9C3 * 10C2 = number of ways to get 3 blue and 2 non-blue<BR>\n" );
document.write( "9C4 * 10C1 = number of ways to get 4 blue and 1 non-blue<BR>\n" );
document.write( "9C5 * 10C0 = number of ways to get 5 blue and 0 non-blue\r<BR>\n" );
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document.write( "Each row is of the form (mCn)*(pCq)<BR>\n" );
document.write( "we have these properties<BR>\n" );
document.write( "m+p = 9 blue + 10 nonblue = 19 marbles total<BR>\n" );
document.write( "n+q = 5 selections allowed\r<BR>\n" );
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document.write( "Calculator verification using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.wolframalpha.com/input?i=%28%289choose2%29+*+%2810choose3%29%29+%2B+%28%289choose3%29+*+%2810choose2%29%29+%2B+%28%289choose4%29+*+%2810choose1%29%29+%2B+%28%289choose5%29%29\">https://www.wolframalpha.com/input?i=%28%289choose2%29+*+%2810choose3%29%29+%2B+%28%289choose3%29+*+%2810choose2%29%29+%2B+%28%289choose4%29+*+%2810choose1%29%29+%2B+%28%289choose5%29%29</a><BR>\n" );
document.write( "The input was \"((9choose2) * (10choose3)) + ((9choose3) * (10choose2)) + ((9choose4) * (10choose1)) + ((9choose5))\" without quotes.\r<BR>\n" );
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document.write( "The \"choose\" function is another name for the nCr combination function.\r<BR>\n" );
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document.write( "Another approach:\r<BR>\n" );
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document.write( "There are 19C5 = 11628 ways to pick 5 marbles of any color. Order doesn't matter.\r<BR>\n" );
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document.write( "Also we have 10C5 = 252 ways to pick five non-blue marbles<BR>\n" );
document.write( "And (9C1)*(10C4) = 9*210 = 1890 ways to pick one blue marble and four non-blue marbles.\r<BR>\n" );
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document.write( "Therefore, we have 11628 - (252+1890) = <font color=red>9486</font> ways to select at least two blue marbles in a batch of five.\r<BR>\n" );
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document.write( "I'm using the idea that<BR>\n" );
document.write( "<font color=blue>n(0 blue) + n(1 blue)</font> + <font color=red>n(at least 2 blue)</font> = n(total)<BR>\n" );
document.write( "which leads to<BR>\n" );
document.write( "<font color=red>n(at least 2 blue)</font> = n(total) - ( <font color=blue>n(0 blue) + n(1 blue)</font> )<BR>\n" );
document.write( "<font color=red>n(at least 2 blue)</font> = 19C5 - ( <font color=blue>10C5 + 9C1*10C4</font> )<BR>\n" );
document.write( "<font color=red>n(at least 2 blue)</font> = 11628 - ( <font color=blue>252 + 1890 </font> )<BR>\n" );
document.write( "<font color=red>n(at least 2 blue)</font> = <font color=red>9486</font>\r<BR>\n" );
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document.write( "Calculator verification using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.wolframalpha.com/input?i=%2819+choose+5%29+-+%2810+choose+5%29+-+%289+choose+1%29*%2810+choose+4%29\">https://www.wolframalpha.com/input?i=%2819+choose+5%29+-+%2810+choose+5%29+-+%289+choose+1%29*%2810+choose+4%29</a><BR>\n" );
document.write( "The input was: (19 choose 5) - (10 choose 5) - (9 choose 1)*(10 choose 4)\r<BR>\n" );
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document.write( "Another relevant calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html\">https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html</a><BR>\n" );
document.write( "It not only computes the nCr value, but it also lists out the various combinations in the \"List Them\" section.<BR>\n" );
document.write( "I recommend exploring with small sets because listing tens of thousands of items would be a tedious task to verify.<BR>\n" );
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