Question

find how many ways the integers 1,2,3,4,5,6,7,8 can be placed in a circle if:

- Atleast three odd numbers are together?

Answer 1

I get 2304

Answer 2

\[(4!)^2+4\times3\times2\times4\times3\times3!=(4!)^2\times 4=2304\]

Answer 3

why do you times it by 4 for?

Answer 4

\[(4!)^2+4\times3\times2\times4\times3\times3!\]
\[=(4!)^2+(4\times3\times2)\times4\times3!\times3\]

\[=(4!)^2+4!\times4!\times3=(4!)^2+(4!)^2\times3=(4!)^2[1+3]\]

\[=(4!)^2\times 4=2304\]