document.write( "Question 1013622: In how many ways can 5 men and 2 women sit in a row if no women sit next to each other? \n" ); document.write( "

Algebra.Com's Answer #630432 by KMST(5328) $\"\"$ $\"About$

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There are $\"7%21=7%2A6%2A5%2A4%2A3%2A2%2A1=42%2A5%21=42%2A120\"$ ways to sit $\"5%2B2=7\"$ people in a row.
\n" ); document.write( "However, some of those arrangements will have 2 women together.
\n" ); document.write( "The two women could be together starting at the first/leftmost position,
\n" ); document.write( "or starting at position number 2,
\n" ); document.write( "or starting at position number 3, or 4, or 5, or 6.
\n" ); document.write( "That is $\"6\"$ positions where the 2 women can be included next to each other.
\n" ); document.write( "For each of those options there are $\"2\"$ choices of which woman is furthest to the left,
\n" ); document.write( "and when all that is decided, there is still the question of how to arrange the men,
\n" ); document.write( "and there are, of course, $\"5%21=4%2A4%2A3%2A2%2A1=120\"$ ways to do that.
\n" ); document.write( "In all there are $\"6%2A2%2A120=12%2A120\"$ ways in which we could have 5 men and 2 women sit in a row with the women next to each other,
\n" ); document.write( "out of the $\"42%2A120\"$ ways that 5 men and 2 women can sit in a row.
\n" ); document.write( "So, the ways that 5 men and 2 women can sit in a row with no women sitting next to each other are
\n" ); document.write( " $\"42%2A120-12%2A120=%2842-12%29%2A120=30%2A120=highlight%283600%29\"$ \n" ); document.write( "

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