I came across this question where I was asked to calculate the probability of drawing 2 identical socks from a drawer containing 24 socks (7 black, 8 blue, and 9 green). When solving this question, I tried tackling it using the counting principle directly as opposed to simply applying the combinatorics formula in hopes to get comfortable with that principle. Though, when I looked up the solution at the end of the book, the answer was completely different, and I am not sure why that is.

What I did was as follows: First, I noticed that when we draw the first sock, we have 24 different options to choose from and then only 1 in order to match it with the first one we chose. Then, I divided it by the total number of ways we can draw 2 socks out of 24 and got the expression:

$\frac{24\cdot 1}{24\cdot 23}=\frac{1}{23}$

The solution at the back of the book took care of each case separately (choosing 2 out of 7 blue socks, OR choosing 2 out of 8 blue socks OR choosing 2 out of 9 green socks), which makes sense, but what is wrong with my approach?

What I did was as follows: First, I noticed that when we draw the first sock, we have 24 different options to choose from and then only 1 in order to match it with the first one we chose. Then, I divided it by the total number of ways we can draw 2 socks out of 24 and got the expression:

$\frac{24\cdot 1}{24\cdot 23}=\frac{1}{23}$

The solution at the back of the book took care of each case separately (choosing 2 out of 7 blue socks, OR choosing 2 out of 8 blue socks OR choosing 2 out of 9 green socks), which makes sense, but what is wrong with my approach?