a grouping problem

Problem: how many ways to group 7 people into three groups,  with 3 people in one group and two people in each of the other two groups? (the two-people groups are not distinguishable)

 

Solution #1:

First select 3 people from the total of 7, there is (7 3) = 7*6*5/(3*2*1) = 35 choices. Then select 2 people from the remaining 4 people, there is (4 2) = 4*3/(2*1)=6 choices. But the two-people grous are indistinguishable, so there is 35 * 6 / 2 = 105 choices.

 

Solution #2:

Use e.g.f: g(x,3) = (x^2/2! + x^3/3!)^3 / 3!, and find the coefficient of x^7/7! is 105.