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How many ways are there for 10 women and six men to stand in a line so that no two men stand next to each other? [Hint: First position the women and then consider possible positions for the men.]

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closed as off-topic by José Carlos Santos, Lord Shark the Unknown, blub, Paul Frost, Javi Apr 19 at 11:13

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    $\begingroup$ That's a great hint in your Question. What did you try? $\endgroup$ – hardmath Dec 1 '13 at 13:28
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As your hint suggests, for this sort of problem, first consider the positions of the women, then the positions of the men.

How many possible ways are there to arrange ten women in a row? It will be $$^{10}P_{10}=10! = 3628800.$$

Now since we need for no two men to stand next to one another, we can visualize the situation as follows

                   * W * W * W * W * W * W * W * W * W * W *

A man can be inserted in any one of the eleven positions marked off with a '*', and this will ensure no two men stand next to each other. Hence, we need to find how many ways we can arrange $6$ men in the $11$ possible places: this is given by $$ ^{11}P_6 = 11\times 10\times 9\times 8 \times 7\times 6 = 332640.$$

Now applying a fundamental law of counting (specifically, the rule of the product), to calculate the total number of possible arrangements satisfying both constraints. This gives us : $$3628800\times 332640 = 1,207,084,032,000$$ which is your required/desired answer.

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Use your hint and think how many positions can be made if you make space between every two women in the line. Then from these spaces,choose 6 of them,and then find in how many ways the guys can be placed in this positions

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Think by using your hint and allowing men and women to be placed in spaces along the line.

Also try to begin by placing 6 men and 6 women firstly.

You will get something like the configuration:

W,M,W,M,W,M,W,M,W,M,W,M

Now think about how could you put the rest of women in that line, where they could enter?

To finish, you have to consider that you can look at that line starting from either a man as a women. (Just look that the line can be read from right to left or from left to right).

Maybe instead of placing one by one like you wanted, you could also think about placing the women first and then looking for a spot where to put a man.

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