Permutation Problem

Question

Three boys and three girls are standing in a single file line. Find the probability that the boys and girls will alternate.

Answer 1

The number of possible arrangements, without gender restrictions, is 6!

 

Lets see how many ways we can do the BGBGBG arrangement.  First we have 3 boys possible for position 1, then two boys for position three, and one remaining boy for position five. That's 3! ways to establish the boys. Then there are, similarly, 3! ways to arrange the girls within this overall pattern.  So it's (3!)² ways. Then there's only one other pattern: GBGBGB, and this has (3!)² ways, too.

 

The desired probability is therefore [2•(3!)² ] / 6! and I leave the computation to you.