Form groups of people using combinatorics I have this problem: 

How many groups of 5 people can be formed between 4 boys and 7 girls
  if there should be at least 2 girls included?

My attempt was:
Ways of select two girls of a total of $7$ is: $\binom{7}{2} = 21$
Ways to select the $3$ remaining people, out of a total of $4$ boys and $5$ girls, is: $\binom{9}{3} = 84$
Therefore, $21 \cdot 84 = 1764$, but according to the guide my answer is incorrect. 
What is wrong with my development? Thanks in advance. 
 A: You are overcounting. 
Let girls be $G_1,G_2,...,G_7$ and boys be $B_1,B_2,B_3,B_4$. Then, you are choosing $2$ girls in the beginning, let's say they are $G_1$ and $G_2$. Then from the rest of $9$, you are choosing $3$ people. Here, suppose you choose $G_3,B_1,B_2$. But this is same as choosing $G_2,G_3$ initially and then choosing $G_1,B_1,B_2$ and also same as choosing $G_1,G_3$ initially and then $G_2,B_1,B_2$. In this case, you counted this specific group $3$ times for instance. 
As an additional note, when you have $4$ girls in group, you will be counting each of those  groups $6$ times and when you have $5$ girls in group, it becomes $10$. So, I cannot find a way to proceed from where you started. If the overcounting amount was same for all the groups you form, we could have divided the result by that amount to get rid of the overcounting but this is not the case here.
You can use case distinction in number of girls in order to solve this problem. As a small hint, since there are only $4$ boys, you have only one case where group contains less than $2$ girls. So you can find the number of complementary cases easily.
A: There are four cases, with 2, 3, 4, or 5 girls in the 5-group.
The solution is: 
$$ {4 \choose 3}{7 \choose 2} + 
  {4 \choose 2}{7 \choose 3} + 
   {4 \choose 1}{7 \choose 4} + 
  {4 \choose 0}{7 \choose 5} = 455 
$$
What you have counted is :
$$ {4 \choose 3}{7 \choose 2} {\color{red} {2 \choose 2}} + 
   {4 \choose 2}{7 \choose 3} {\color{red} {3 \choose 2}} + 
   {4 \choose 1}{7 \choose 4} {\color{red} {4 \choose 2}} + 
   {4 \choose 0}{7 \choose 5} {\color{red} {5 \choose 2}} = 1764 
$$
meaning that between the choosen girls there are two special, i.e. the two ones selected in the beggining to ensure the quota.
