There are $3$ men and $3$ women.They should be arranged in such a way that no man stands next to another man how many ways is it possible ?
I got as far as – there can either be a man standing first or a woman. so it would be either M1 W1 M2 W2 M3 W3 or W1 M1 W2 M2 W3 M3
for the first possibility,men can be arranged in 3! ways and women can also be arranged in $3!$ ways so they can be arranged in $3!*3!$ ways which is $36$ ways.
This is also the case for the second possibility
so there are $36+36=72$ ways of arranging them but the answer is $144$ ways $(72\times2)$ how???
Best Answer
You should be more careful re headings. Your heading contradicts the actual question !
As for the question, the men can be inserted at any $3$ of the uparrows, $\uparrow W \uparrow W \uparrow W \uparrow$
thus, using permutation notation, # of ways = $^4P_3\times{^3P_3} = 144$