+612 The Smith family has 4 sons and 3 daughters. In how many ways can they be seated in a row of 7 chairs such that at least 2 boys are next to each other?
+4622 The total number of ways of seating \(7\) people is \(7!\)
The only disallowed seating arrangement is alternate boy-girl, \(\texttt{BGBGBGB}.\) There are \(4!\) ways to seat the boys and \(3!\) ways to seat the girls. So the number of disallowed seating arrangements is \(4! \times 3!\)
The final answer is \(7! - 4! \times 3! = 5040-24\times 6=5040-144=4896\)
+4622 The total number of ways of seating \(7\) people is \(7!\)
The only disallowed seating arrangement is alternate boy-girl, \(\texttt{BGBGBGB}.\) There are \(4!\) ways to seat the boys and \(3!\) ways to seat the girls. So the number of disallowed seating arrangements is \(4! \times 3!\)
The final answer is \(7! - 4! \times 3! = 5040-24\times 6=5040-144=4896\)
