Suppose three men and three women are standing in a line. In how many ways can they be arranged so that every man is standing next to a woman? Let the men and women be labeeled M and W respectively.
MWMWMW - transpose any man and woman next to each other
WMMWMW
MWWMMW
MWMWWM
Interchange M's and W.s
WMWMWM - transpose any man and woman next to each other
MWWMWM
WMMWWM
WMWMMW
We cannot have three men standing next to each other because the middle man would not be standing next to a woman, and we cannot have two men standing at the end, because the man at the end would not be standing next to a woman.
There are eight possibilities so far, but all the men are individuals and so are all the women. Both of these introduce factrors of 3! to take account of the fact that 3 individuals can be arranged in 3! ways.
There are
\[8 \times 3! \times 3!=288\]
ways.