Let $$n$$ be the number of ways in which $$20$$ identical balloons can be distributed among $$5$$ girls and $$3$$ boys such that everyone gets at least one balloon and no girl gets fewer balloons than a boy does. Then

Let's start by taking cases for the number of balloons a boy can get. We'll consider the following cases:

• The maximum balloons any boy gets is 1: In this case, all boys will have to have at least one balloon (1 way) and there will be $$17$$ balloons that will be distributed among girls in $$^{17-1}C_{5-1} = ^{16}C_4 = 1820$$ ways.
  
• The maximum balloons any boy gets is 2: In this case, we can have three sub-cases, the first where only one boy gets two balloons (3 ways), the second where two boys get two balloons and the other gets one (3 ways), and the third where all boys get 2 balloons (1 way). Keeping in mind that no girl gets fewer balloons than any boy, we get the total number of cases as $$3*{}^{6+5-1}C_{5-1} + 3* {}^{5+5-1}C_{5-1} + 1* {}^{4+5-1}C_{5-1} = 630+378+70= 1078$$
• The maximum balloons any boy gets is 3: In this case, there is only one possibility. One of the three boys should get 3 balloons, other two should get 1 balloon each, and all the 5 girls get 3 balloons each as well. This amounts to a total of 3 possibilities.

The total number of possibilities, therefore, are 1820+1078+3=2901

The correct answer is $$n<7000$$