Combinatorics
In how many ways can three boys and three girls sit in a row if they must alternate?
Answer
There are 2 possible alternating patterns:
\[
\text{(B G B G B G)} \quad \text{or} \quad \text{(G B G B G B)}.
\]
In each pattern, we have 3 distinct seats for boys and 3 distinct seats for girls.
\[
\text{Ways to arrange boys in their 3 positions}
= 3! = 6,
\quad
\text{and similarly for girls}
= 3! = 6.
\]
Hence, for one pattern:
\[
3! \times 3! = 36 \text{ ways}.
\]
Since there are 2 patterns, the total number of ways is:
\[
2 \times 36 = 72.
\]
Notes and comments
Comment 1: People are distinguishable, so when talking about arranging people, the order of individuals matters, hence permutations.