Question 1034761
How many ways 10 hats of different colours can be put on 
the heads of 10 boys sitting in a row,so that the red and 
black coloured hats can never be put on the heads of any 
two adjacent boys ?
<b><pre>
The answer would be 10! if there were no restrictions.

If the hats were:

A B C D E F G H I J

and there were no restrictions they could be arranged in 10! ways

From that 10! we must subtract the number of non-allowable ways.

Suppose the red hat is hat A and the black hat is hat B.

Then we must subtract the 9! ways they could be
arranged with hat A and hat B together, with hat A 
left of B:

AB C D E F G H I J

And we must also subtract the 9! ways they could be
arranged with hat A and hat B together, with hat B left 
of hat A:

BA C D E F G H I J

Answer: 10! - 2*9! = 3628800 - 2*(362880) = 2903040

Edwin</pre></b>