Lesson Men and women standing in line
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<H2>Men and women standing in line</H2> <H3>Problem 1</H3>Eight men and five women are standing in a line. In how many arrangements will no two women be standing next to one another? <B>Solution</B> <pre> Imagine that the 8 men are standing in line with intervals (gaps) between them, as shown in the Figure below. * M * M * M * M * M * M * M * M * (" * "s show intervals, or gaps). You see 7 stars between "M"s and two stars before and after the M-line; in all, 7+2 = 9 stars. These stars are the only places for women: Each woman can occupy one and only one position at the star place. So, we can position 8 men by 8! = 8*7*6*5*4*3*2*1 = 40320 ways. And we can place first woman in any of nine (*)-positions; second woman in any of remaining eight (*)-positions; third woman in any of remaining seven positions; fourth . . . and fifth . . . Thus, in all, there are 8!*9*8*7*6*5 = 40320*9*8*7*6*5 = 609638400 differennt arrangements satisfying given conditions. <U>ANSWER</U> </pre> <H3>Problem 2</H3>Eight women and five men are standing in a line. In how many arrangements will no two men be standing next to one another? <B>Solution</B> <pre> Imagine that the 8 women are standing in line with intervals (gaps) between them, as shown in the Figure below. * W * W * W * W * W * W * W * W * (" * "s show intervals, or gaps). You see 7 stars between "W"s and two stars before and after the W-line; in all, 7+2 = 9 stars. These stars are the only places for men: Each man can occupy one and only one position at the star place. So, we can position 8 women by 8! = 8*7*6*5*4*3*2*1 = 40320 ways. And we can place first man in any of nine (*)-positions; second man in any of remaining eight (*)-positions; third man in any of remaining seven positions; fourth . . . and fifth . . . Thus, in all, there are 8!*9*8*7*6*5 = 40320*9*8*7*6*5 = 609638400 differennt arrangements satisfying given restrictions. <U>ANSWER</U> </pre> Surely, these two problems are TWINS. I placed them both together to highlight the major idea of the solution. We first arrange a greater amount of items in a line. Then we place the other sort of items in the gaps of this line. 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