document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174468>Question 1174468</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, five green ones, two white ones, and two purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms.<BR>\n" );
document.write( "She has no more than one white one. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850656 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $R, G, W, P$ be the number of red, green, white, and purple marbles in the bag, respectively.<BR>\n" );
document.write( "We have $R=3$, $G=5$, $W=2$, $P=2$. The total number of marbles is $3+5+2+2=12$.<BR>\n" );
document.write( "Suzan grabs 8 marbles.<BR>\n" );
document.write( "The total number of ways to grab 8 marbles from 12 is $\binom{12}{8} = \binom{12}{4} = \frac{12 \cdot 11 \cdot 10 \cdot 9}{4 \cdot 3 \cdot 2 \cdot 1} = 495$.\r<BR>\n" );
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document.write( "We want to find the probability that she has no more than one white marble. This means she has either zero white marbles or one white marble.<BR>\n" );
document.write( "Case 1: Zero white marbles.<BR>\n" );
document.write( "If she has zero white marbles, she grabs 8 marbles from the remaining 10 marbles (3 red, 5 green, 2 purple).<BR>\n" );
document.write( "The number of ways to do this is $\binom{10}{8} = \binom{10}{2} = \frac{10 \cdot 9}{2 \cdot 1} = 45$.\r<BR>\n" );
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document.write( "Case 2: One white marble.<BR>\n" );
document.write( "If she has one white marble, she grabs 7 marbles from the remaining 10 marbles.<BR>\n" );
document.write( "The number of ways to do this is $\binom{10}{7} = \binom{10}{3} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120$.<BR>\n" );
document.write( "Since there are 2 white marbles, she can pick one of them in $\binom{2}{1} = 2$ ways.<BR>\n" );
document.write( "So the number of ways to have one white marble is $120 \times 2 = 240$.\r<BR>\n" );
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document.write( "The total number of ways to have no more than one white marble is $45 + 240 = 285$.<BR>\n" );
document.write( "The probability is $\frac{285}{495}$.<BR>\n" );
document.write( "We can simplify this fraction.<BR>\n" );
document.write( "Divide by 5: $\frac{285}{5} = 57$ and $\frac{495}{5} = 99$.<BR>\n" );
document.write( "So we have $\frac{57}{99}$.<BR>\n" );
document.write( "Divide by 3: $\frac{57}{3} = 19$ and $\frac{99}{3} = 33$.<BR>\n" );
document.write( "So we have $\frac{19}{33}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{19}{33}}$<BR>\n" );
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