# In how many ways can 6 graduates line up to receive their diplomas, if there are 4 girls and 2 boys and the girls will receive their diplomas first?

Jul 19, 2016

48

#### Explanation:

this corresponds to how many ways girls can line up and then for each of these ways how we can include the boys lined up. Lets Consider how many ways can 2 people line up...
$p 1 , p 2 \mathmr{and} p 2 , p 1$ thus $f \left(2\right) = 2$

if there are 3 people
$p 1 , p 2 , p 3 \mathmr{and} p 1 , p 3 , p 2 \mathmr{and} p 2 , p 1 , p 3 \mathmr{and} p 2 , p 3 , p 1 \mathmr{and} p 3 , p 1 , p 2 \mathmr{and} p 3 , p 2 , p 1$ thus $f \left(3\right) = 6$

in fact this pattern will repeat in the following ways f(n)=n!

so when the girls line up they will line up 4! ways

when only the boys line up they will do so in 2! ways

so girls can line up $24$ ways followed by two boys in 2 different ways for each of the ways that the girls can line up

$24 \cdot 2 = 48$