document.write( "Question 733245: A committee consisting of at least three members is to be formed from a group of six boys and six girls such that it always has a boy and a girl. Find the number of ways to form such committee. \n" ); document.write( "

Algebra.Com's Answer #448645 by Edwin McCravy(20060) $\"\"$ $\"About$

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A committee consisting of at least three members is to be formed from a group
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document.write( "First we count all committees of any size without regard to size or sex.\r\n" );
document.write( "(This will even include a committee of size 0.)  Then we'll subtract the \r\n" );
document.write( "number of unacceptable committees.\r\n" );
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document.write( "In choosing a committee without regard to size or sex, we \r\n" );
document.write( "tell each of the 12 people one of two things:\r\n" );
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document.write( "1.  YES, you will serve on the committee\r\n" );
document.write( "2.  NO, you will NOT serve on the committee.\r\n" );
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document.write( "If there were no restrictions with regard to size or sex the answer \r\n" );
document.write( "would be 2¹².\r\n" );
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document.write( "Now we subtract the unacceptable committees.\r\n" );
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document.write( "We may not tell all the girls NO, for that would leave an all-boy committee.\r\n" );
document.write( "So that is the number of ways we could tell all the girls NO, and tell the 6\r\n" );
document.write( "boys YES or NO, which would be  2⁶.  \r\n" );
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document.write( "We also may not tell all the boys NO, for that would leave an all-girl\r\n" );
document.write( "committee. So that is the number of ways we could tell all the boys NO, and\r\n" );
document.write( "tell the 6 girls YES or NO, which would also be  2⁶.\r\n" );
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document.write( "That would be 2·2⁶ or 2⁷ to subtract. However both those\r\n" );
document.write( "have 1 case in common, the case where all 12 people are told NO, so we must\r\n" );
document.write( "avoid subtracting that case twice, so we can only subtract 1 less than that or\r\n" );
document.write( "2⁷ - 1  \r\n" );
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document.write( "So without regard to the committee size, there are\r\n" );
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document.write( "2¹² - (2⁷ - 1) or 2¹² - 2⁷ + 1 committees.\r\n" );
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document.write( "Now we only have to subtract the number of committees of size 2 consisting\r\n" );
document.write( "of 1 boy and 1 girl.  There are 6 ways to choose the girl and 6 ways to\r\n" );
document.write( "choose the boy.  That's 6·6 or 36 committees of 2 to subtract.\r\n" );
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document.write( "[Note: We have already subtracted the committees of sizes 0 and 1. The\r\n" );
document.write( "committee of size 0 was the one we had to avoid counting twice above, and a\r\n" );
document.write( "committee of 1 cannot contain both sexes.]\r\n" );
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document.write( "Final answer:  2¹² - 2⁷ + 1 - 36 = 4096-128+1-36 = 3933\r\n" );
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document.write( "Edwin

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