A line of boys and girls

Problem

A class has $4$ boys and $2$ girls. (a) In how many ways can they be lined up? (b) In how many ways if the two girls must stay together? (c) In how many ways if boys and girls stay in the two «blocks» (all the boys together, all the girls together)?

Solution

There are $6$ people in all.

(a) The permutations of $6$ distinct people: $6! = \boxed{720}$

(b) We treat the two girls as a single block: there remain $5$ elements to permute ($5!$), and the two girls swap internally ($2!$): $5!\cdot 2! = 120\cdot 2 = \boxed{240}$

(c) Two blocks (boys, girls) that can be arranged in $2!$ orders; inside the boys’ block there are $4!$ orders, inside the girls’ block $2!$: $4!\cdot 2!\cdot 2! = 24\cdot 2\cdot 2 = \boxed{96}$

Links

Topics: Probabilita
Concepts: Permutazioni
Methods: Fattoriale permutazioni
Skills: Calcolo combinatorio
Exercise type: Problema combinatorio