Please help me in understanding the difference between this two question:
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?