How many ways 10 hats of different colours can be put on the heads of10 boys sitting in a row, so that the red and the black coloured hats cannever be put on the heads of any two adjacent boys?

How many ways 10 hats of different colours can be put on the heads of1

1.	How many ways 10 hats of different colours can be put on the heads of10 boys sitting in a row, so that the red and the black coloured hats cannever be put on the heads of any two adjacent boys?
Answer» 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! - 29! = 3628800 - 2(362880) = 2903040