document.write( "Question 103819: in how many ways can 4 boys and 2 girls be seated if
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Algebra.Com's Answer #75475 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "in how many ways can 4 boys and 2 girls be seated if\r\n" );
document.write( "i)the 2 girls are to seat next to eachother\r\n" );
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document.write( "We have two cases:\r\n" );
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document.write( "1. Girl #1 sits left of Girl #2\r\n" );
document.write( "2. Girl #2 sits left of Girl #1\r\n" );
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document.write( "In case 1, imagine that Girl #1's right wrist is handcuffed to Girl #2's left\r\n" );
document.write( "wrist. Thinking of it that way, we actually only have 5 \"things\" to arrange,\r\n" );
document.write( "namely 4 boys and 1 pair of handcuffed girls.\r\n" );
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document.write( "We can seat the first \"thing\" on the far left any of 5 ways.\r\n" );
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document.write( "For each of those 5 ways that we can seat a \"thing\" on the far left, we can\r\n" );
document.write( "choose any of the remaining 4 things to sit second from the left.  That's 5x4\r\n" );
document.write( "or 20 ways to seat something first and second from the left. \r\n" );
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document.write( "For each of those 5x4 or 20 ways that we can seat two \"things\" on the far left,\r\n" );
document.write( "we can choose any of the remaining 3 things to sit third from the left.  That's\r\n" );
document.write( "5x4x3 or 60 ways to seat something first, second and third from the left.\r\n" );
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document.write( "For each of those 5x4x3 or 60 ways that we can seat three \"things\" on the left,\r\n" );
document.write( "we can choose either of the remaining 2 \"things\" to sit fourth from the left.\r\n" );
document.write( "That's 5x4x3x2 or 120 ways to seat something first, second, third and fourth\r\n" );
document.write( "from the left.\r\n" );
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document.write( "For each of those 5x4x3x2 or 120 ways that we can seat four \"things\" on the\r\n" );
document.write( "left, we can choose only the 1 remaining \"thing\" to sit fifth from the left, or\r\n" );
document.write( "on the far right.  That's 5x4x3x2x1 or 120 ways to seat something first,\r\n" );
document.write( "second, third, fourth and fifth from the left.  Now all are seated.\r\n" );
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document.write( "So there are 5x4x3x2x1 or 120 ways for case 1.\r\n" );
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document.write( "Now if we imagine handcuffing the two girls the other way, that is, imagine\r\n" );
document.write( "that Girl #2's right wrist is handcuffed to Girl #1's left wrist.  Then we get\r\n" );
document.write( "exactly the same number 120 again.\r\n" );
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document.write( "So putting the two 120's together, we get 240 ways of seating them so that the\r\n" );
document.write( "two girls sit side by side. \r\n" );
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document.write( "This can also be thought of as\r\n" );
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document.write( "1. There are 2P2 of 2! or 2x1 or 2 ways to handcuff the two girls.\r\n" );
document.write( "2. For each of the 2P2 or 2! or 2 way to handcuff the two girls, there are \r\n" );
document.write( "   5P5 or 5! or 5x4x3x2x1 or 120 ways to seat the 5 \"things\".\r\n" );
document.write( "3. So the result can be written as (2P2)(5P5) or (2!)(5!) or (2)(120) = 240.\r\n" );
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document.write( "ii)the 2 girls are not to seat next to each other.\r\n" );
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document.write( "Here the strategy is \r\n" );
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document.write( "1. First we determine the number of ways any of the 6 can sit anywhere.\r\n" );
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document.write( "then\r\n" );
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document.write( "2. Subtract the 240 ways from part (i), which will remove all cases where\r\n" );
document.write( "the girls sit together.\r\n" );
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document.write( "First we calculate the number of ways any of the 6 can sit anywhere:\r\n" );
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document.write( "We can seat the first person on the far left any of 6 ways.\r\n" );
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document.write( "For each of those 6 ways that we can seat a person in the far left seat, we can\r\n" );
document.write( "choose any of the remaining 5 people to sit in the second seat from the left. \r\n" );
document.write( "That's 6x5 or 30 ways to seat somebody in the first two seats on the left. \r\n" );
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document.write( "For each of those 6x5 or 30 ways that we can seat somebody in the two leftmost\r\n" );
document.write( "seats, we can choose any of the remaining 4 people to sit in the third seat\r\n" );
document.write( "from the left.  That's 6x5x4 or 120 ways to seat somebody in the first three\r\n" );
document.write( "seats on the left.\r\n" );
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document.write( "For each of those 6x5x4 or 120 ways that we can seat somebody in the three\r\n" );
document.write( "leftmost seats, we can choose any of the remaining 3 people to sit in the\r\n" );
document.write( "fourth seat from the left.  That's 6x5x4x3 or 360 ways to seat somebody in the\r\n" );
document.write( "first four seats on the left.\r\n" );
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document.write( "For each of those 6x5x4x3 or 360 ways that we can seat somebody in the four\r\n" );
document.write( "leftmost seats, we can choose any of the remaining 2 people to sit in the fifth\r\n" );
document.write( "seat from the left.  That's 6x5x4x3x2 or 720 ways to seat somebody in the first\r\n" );
document.write( "five seats on the left.\r\n" );
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document.write( "For each of those 6x5x4x3x2 or 720 ways that we can seat somebody in the fifth\r\n" );
document.write( "leftmost seats, we can only choose the one remaining person to sit in the sixth\r\n" );
document.write( "seat from the left, or the last seat on the right.  That's 6x5x4x3x2x1 or 720\r\n" );
document.write( "ways to seat somebody in all 6 seats in every possible order.\r\n" );
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document.write( "So the final answer will be these 720 MINUS the 240 ways from part (i) where \r\n" );
document.write( "the two girls sit together, and we get 720 - 240 or 480 ways when the girls\r\n" );
document.write( "do not sit together.\r\n" );
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document.write( "The results of this problem can also be thought of and written either in terms\r\n" );
document.write( "of permutations or factorials:\r\n" );
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document.write( "We can arrange the 6 people 6P6 or 6! or 720 ways.\r\n" );
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document.write( "We can \"handcuff\" the two girls 2! or 2 ways.\r\n" );
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document.write( "We can seat the 5 things (4 boys and a pair of handcuff girls) 5P5 or 5! or 120 ways.\r\n" );
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document.write( "So, if you prefer, in terms of permutations or factorials, we have:\r\n" );
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document.write( "6P6 - (2P2)(5P5) = 6! - (2!)(5!) = 720 - 2(120) = 720 - 240 = 480. \r\n" );
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document.write( "Edwin
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