document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193951>Question 1193951</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A binary string is an ordered sequences of 0's and 1's (for example, 0100101). Determine the number of binary strings there are consisting of five 0's and three 1's. \r<BR>\n" );
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document.write( "Textbook answer is 56.<BR>\n" );
document.write( "Can someone help me solve this problem? Thanks. \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 #826018 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your solution here will either be 8C5 of 8C3.<BR>\n" );
document.write( "your string is 8 digits long.\r<BR>\n" );
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document.write( "8C5 is the number of ways you can get a string of 5 zeroes out of a string of 8 digits that that has 5 zeroes in it.  the value of the other 3 digits is inconsequential since they just fill the spaces that don't contain zeroes.\r<BR>\n" );
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document.write( "8C3 is the number of ways you can get a string of 3 ones out of a string of 8 digits that has 3 ones in it.  the value of the other 5 digits is inconsequential since they just fill the spaces that don't contain ones.\r<BR>\n" );
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document.write( "a simple example is a string of 3 digits that contains 2 zeroes and 1 one.<BR>\n" );
document.write( "the number of ways of getting 2 zeroes and 1 one would be either 3C2 or 3C1.<BR>\n" );
document.write( "3C2 = 3*2/2 = 3<BR>\n" );
document.write( "3C1 = 3/1 = 3<BR>\n" );
document.write( "there are 3 ways you can get a string of 2 zeroes and 1 one.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "001<BR>\n" );
document.write( "010<BR>\n" );
document.write( "100\r<BR>\n" );
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document.write( "the formula of 8C5 is also shown as C(8,5).<BR>\n" );
document.write( "it is equal to 8! / (5! * 3!) which is equal to (8 * 7 * 6 * 5! / (5! * 3!) which is equal to (8 * 7 * 6 / (3 * 2 * 1) which is equal to 56.\r<BR>\n" );
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document.write( "it is a combination formula, not  a permutation formula.<BR>\n" );
document.write( "normally you would use a permutation formula for an ordered set, but in this case, the different sets would be equivalent, as shown below in the simple example.\r<BR>\n" );
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document.write( "assume the 3 digits are a,b,c\r<BR>\n" );
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document.write( "the number of possible permutations are:<BR>\n" );
document.write( "abc<BR>\n" );
document.write( "acb<BR>\n" );
document.write( "bac<BR>\n" );
document.write( "bca<BR>\n" );
document.write( "cab<BR>\n" );
document.write( "cba\r<BR>\n" );
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document.write( "since a and b are equal to 0 and c is equal to 1, this becomes:\r<BR>\n" );
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document.write( "001<BR>\n" );
document.write( "010<BR>\n" );
document.write( "001<BR>\n" );
document.write( "010<BR>\n" );
document.write( "100<BR>\n" );
document.write( "100\r<BR>\n" );
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document.write( "3 of the sets are duplicates of the other 3 sets, so the number of possible sets that are different from each other becomes 3.\r<BR>\n" );
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document.write( "3P2 is equal to 3! / 1! = 6<BR>\n" );
document.write( "3C2 is equal to 3! / (1! * 2!) = 3<BR>\n" );
document.write( "3P2 gives you 6 sets where order is important, but since the individual values are not all different, then 3P2 reduces to 3C2.<BR>\n" );
document.write( "that's why the combination formula works in this case, where it wouldn't if the individual values were all different from each other.\r<BR>\n" );
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document.write( "bottom line:<BR>\n" );
document.write( "your answer is either 8C5 or 8C3.<BR>\n" );
document.write( "8C5 is the same as C(8,5)<BR>\n" );
document.write( "3C3 is the same as C(8,3)<BR>\n" );
document.write( "8C5 or C(8,5) is equal to 8! / (5! * 3!)<BR>\n" );
document.write( "8C3 or C(8,3) is equal to 8! / (3! * 5!)<BR>\n" );
document.write( "as you can calculate, they will both give you the same answer as 56.\r<BR>\n" );
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document.write( "let me know if you have any questions.<BR>\n" );
document.write( "theo\r<BR>\n" );
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