SOLUTION: 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 adja

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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 ?
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
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 ?
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
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