Question 1108560
If we start by selecting and setting aside what the third and fourth children will receive,
we have {{{7}}} choices for our first pick (to be set into a bag labeled for child # 4),
and then we have {{{6}}} choices for our second pick (to be reserved for child # 3.
After that, we have {{{7-1-1=5}}} balloons left.
There are {{{5*4/2=10}}} different ways to pick the 2 balloons to be given to child # 2.
Our choices multiply, so we had
{{{7*6*10=highlight(420)}}} different ways to decide who gets what balloon or what set of balloons.