document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=323342>Question 323342</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box contains ten balls numbered from 1 to 10. If you pick two balls at random, what is the probability that the sum of the numbers on the two balls is even?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #231442 by </B><A HREF=/tutors/aboutme.mpl?userid=galactus><B>galactus(183)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=galactus><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/cgi-bin/show-face.mpl?user=galactus><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=231442\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are 2 balls chosen. Thus, there are 10^2 = 100 possible outcomes.\r<BR>\n" );
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document.write( "Half of them sum to odds and half to evens.\r<BR>\n" );
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document.write( "Perhaps you have not dealt with generating functions, but we can illustrate it with them.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sum%28x%5Ek%2Ck=1%2C10%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sum%28x%5Ek%2Ck=1%2C10%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we expand this out, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E20%2B2x%5E19%2B3x%5E18%2B4x%5E17%2B5x%5E16%2B6x%5E15%2B7x%5E14%2B8x%5E13%2B9x%5E12%2B10x%5E11%2B9x%5E10%2B8x%5E9%2B7x%5E8%2B6x%5E7%2B5x%5E6%2B4x%5E5%2B3x%5E4%2B2x%5E3%2Bx%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The coefficients represent the number of ways that each sum can occur. i.e. how many ways can we get a sum of 15?. Look at the coefficient of the x^15 term.<BR>\n" );
document.write( "It is 6. There are 6 ways to get a sum of 15. How many ways can we get a sum of 20?. Look at x^20 and we see its coefficient is 1. There is 1 way to get a sum of 20, and that is by drawing two 10's.<BR>\n" );
document.write( "Since there are 100 possible outcomes, count up the coefficients of the even exponents. They sum to 50. Therefore, the probability is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2F100=1%2F2 ALIGN=MIDDLE  ALT=\"50%2F100=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the two balls summing to an even number. The other half sums to odd numbers. </SPAN>\n" );
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