There are $7$ boys and $6$ girls in a group. $2$ of boys and $2$ of girls have glasses.What is the probability that $2$ people have glasses or girls. There are $7$ boys and $6$ girls in a group. $2$ of boys and $2$ of girls have glasses.That is, $5$ boys and $4$ girls do not have glasses.What is the probability that choosing $2$ people from the group have glasses or girls.
My attempt : All conditions $C(13,2)=78$ ,being girls = $C(6,2)=15$ , having glasses = $C(4,2)=6$
so my answer was $(15+6=21)/78$ .
However , my book says that the answer is $28/78$ , it says that the numerator is equal to $C(8,2)=28$
Note: In my answer , it seems that there is overcounting , and i should subtract $C(2,2)=1$ , but the cases are separate..
Thus , i wonder that why my solution is wrong.
 A: First find the subset of individuals in the group that either have glasses or are girls. Notice that to find this subset we have to take a Union between individuals with glasses and individuals who are girls. From the problem it says there are ${4}$ People with Glasses and ${6}$ Girls. However, there is an intersection between these sets -- two of the girls also have glasses so if you just say ${6+4}$ you've double counted ${2}$ girls. Thus the Union of the set is ${6 + 4 - 2 = 8}$.
Now you can choose ${2}$ people from the subset of ${8}$ in ${C(8,2)}$ ways. There are ${13}$ total individuals in the group so you can randomly choose ${2}$ in ${C(13,2)}$ ways. The probability then is ${\frac{C(8,2)}{C(13,2)}}$
A: I suspect you are translating from anther language, since "choosing 2 people from the group have glasses or girls," isn't really English.  The problem is whether the question means "What is the probability that both of them wear glasses or both of them are girls," which is the question you answered, or whether it means "What is the probability that each of them is either a girl or wears glasses," which is the question the book is answering.
Note that this is the same as "What is the probability that each of the is either a girl or a boy with glasses," so there are $8$ kids to choose from.
You obviously understand the math involved, which is what's important.
