# Understanding differences

How many ways can $4$ men and $4$ women stand in a line if the women and men alternate ?

How many ways can $4$ men and $4$ women stand in a line if no two women are together ?

We employ a bit of different strategies for solving them and the answer is also different,but I am not understanding why is this differences.Kindly explain.

I am in bit of a doubt in another similar problem:

Find the number of ways of arranging $21$ white balls and $3$ black balls in a row so that no two black balls are together.

I think the answer should be $21! \times ^{24}P_3$ but this is incorrect.Could somebody explain?

• By multiplying by $21!$, and by using $P$ instead of $C$, you're assuming you can tell the difference beween the white balls, and can tell the difference between the black balls. (i.e. B1 B2 W1 W2... is different from B2 B1 W1 W2...) Usually when dealing with balls, they're indistinguishable.
• The answer $^{24}C_3$ (i.e. choosing any 3 balls out of the 24 to be black) would be incorrect because it doesn't exclude answers such as BWBBW... where two black balls appear next to each other.
• Thanks,now I understand where I was wrong,I realized that the answer should be $^{22}C_3$.A possible explanation is: If we arrange the 21 black balls first then in order to have a white ball between two black balls we have 22 cases. Dec 21, 2010 at 14:18