There are 9 color balls at a row. From that how many ways can 9 different color balls be arranged in a row so that black, white, red and green are never together?
Choose the correct choice from the following answers:
A) 223600 B) 214600 C) 566600 D) 786789 E) 345600
Total number of ways in which 9 different colour balls can be arranged in a row   =9! ⋯(A)   Now we will find out total number of ways in which 9 different colour balls can be arranged in a row so that black, white, red and green balls are always together.   We have total 9 balls. Since black, white, red and green balls are always together, group these 4 balls together and consider as a single ball. Hence we can take total number of balls as 6. These 6 balls can be arranged in 6! ways.   We had grouped 4 balls together. These 4 balls can be arranged among themselves in 4! ways.   Hence, total number of ways in which 9 different colour balls be arranged in a row so that black, white, red and green balls are always together   =6!×4! ⋯(B)   From (A) and (B),   Total number of ways in which 9 different colour balls can be arranged in a row so that black, white, red and green balls are never together   =9!–6!×4!=6!×7×8×9−6!×4!=6!(7×8×9–4!)=6!(504–24)=6!×480=720×480=345600   345600
<result>E</result>
There are 9 color balls at a row. From that how many ways can 9 different color balls be arranged in a row so that black, white, red and green are never together?