document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210234>Question 1210234</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors?<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 #851708 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=851708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> That's a colorful sock drawer! Let's figure out the minimum number of socks Vera needs to pull to guarantee six matching pairs of different colors.\r<BR>\n" );
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document.write( "To ensure she has six matching pairs of *different* colors, we need to consider the worst-case scenario. Imagine Vera keeps picking one sock of each color before she gets a second one of any color.\r<BR>\n" );
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document.write( "Here's the breakdown:\r<BR>\n" );
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document.write( "1.  She could pick one white, one black, one brown, one blue, one red, and one green sock. That's 6 socks, and no pairs yet.<BR>\n" );
document.write( "2.  She could then pick a second white sock, making one white pair. That's 7 socks.<BR>\n" );
document.write( "3.  Next, she could pick a second black sock, making one black pair. That's 8 socks.<BR>\n" );
document.write( "4.  She could continue this pattern, picking one of each of the remaining colors to form a pair. After picking a second brown, blue, red, and green sock, she would have 6 pairs of different colors. This would be $6 + 6 = 12$ socks.\r<BR>\n" );
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document.write( "However, we need to *ensure* at least six matching pairs of *different* colors. Consider the absolute worst luck:\r<BR>\n" );
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document.write( "* She could pick all 20 white socks.<BR>\n" );
document.write( "* Then, she could pick all 21 black socks.<BR>\n" );
document.write( "* Then, she could pick all 22 brown socks.<BR>\n" );
document.write( "* Then, she could pick all 23 blue socks.<BR>\n" );
document.write( "* Then, she could pick all 24 red socks.\r<BR>\n" );
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document.write( "At this point, she has $20 + 21 + 22 + 23 + 24 = 110$ socks, and she still doesn't have six matching pairs of *different* colors. The very next sock she picks *must* be green, creating her first green pair.\r<BR>\n" );
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document.write( "Now, to guarantee six different colored pairs, after picking all of the socks of five colors, she would need to pick two socks of the sixth color.\r<BR>\n" );
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document.write( "So, the worst-case scenario to guarantee six matching pairs of different colors is:\r<BR>\n" );
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document.write( "* Pick all the socks of 5 colors (the largest quantities): $25 + 24 + 23 + 22 + 21 = 115$ socks.<BR>\n" );
document.write( "* Then, pick 2 socks of the remaining color (white) to form a sixth pair.\r<BR>\n" );
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document.write( "Therefore, Vera must pull out a minimum of $115 + 2 = 117$ socks to ensure at least six matching pairs of different colors. </SPAN>\n" );
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