Hello, mauro21pl!

Quote:

a) In how many ways can 3 boys and 3 girls sit in a row?

With six people, there are: .$\displaystyle 6! = 720$ ways.

Quote:

b) In how many ways can 3 boys and 3 girls sit in the row

if the boys and the girls are each to sit together?

There are two arrangements of the genders: .$\displaystyle BBBGGG$ or $\displaystyle GGGBBB$

In each, the three boys can be ordered in $\displaystyle 3!$ ways

. . and the girls can be ordered in $\displaystyle 3!$ ways.

Therefore, there are: .$\displaystyle 2 \times 3! \times 3! \:=\;72$ ways.

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c) In how many ways if only the boys must sit together?

Duct-tape the boys together.

Then there are four "people" to arrange. .There are $\displaystyle 4!$ arrangements.

For each arrangement, the three boys can be ordered in $\displaystyle 3!$ ways.

Therefore, there are: .$\displaystyle 4! \times 3! \:=\:144$ ways.

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d) In how many ways if no two people of the same sex are allowed to sit together?

Since the boys and girls must alternate,

. . there are two arrangements: .$\displaystyle BGBGBG$ or $\displaystyle GBGBGB$

For each arrangement, the three boys can be ordered in $\displaystyle 3!$ ways

. . and the three girls can be ordered in $\displaystyle 3!$ ways.

Therefore, there are: .$\displaystyle 2 \times 3! \times 3! \:=\:72$ ways.