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# help me.. thank you!

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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?

Apr 21, 2018

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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 $$7! - 4! \times 3! = 5040-24\times 6=5040-144=4896$$

.
Apr 21, 2018

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 $$7! - 4! \times 3! = 5040-24\times 6=5040-144=4896$$