Probability of having at least one pair by drawing 4 shoes from 12 pairs. There are $12$ pairs of shoes in a cupboard. $4$ are drawn at random. What is the probability that there is at least one pair?

My first attempt: If we chose a pair at first and then draw any two at random from the rest, then there will be at least one pair. 
We can choose one pair in ${12}\choose 4$ ways and chose any $2$ from the rest in ${22}\choose 2$ ways.
Therefore, the required probability= $\frac{{12\choose 4} \times {22\choose 2}}{{24\choose 4}}$ = $\frac{6}{23} =\frac{42}{161}$ 

But the given answer is $\frac{41}{161}$.

Another attempt: Each of the 4 shoes we choose, will come from one of the pairs. We can choose the four pairs in ${12\choose 4}$ ways and can select a shoe from each of the pairs in $2$ ways so that no pair is obtained. Therefore, required probability =$1-$ $\frac{{12\choose 4} \times 2^4}{{24\choose 4}}$ = $\frac{41}{161}$

What is wrong with the first attempt?
Calculate $1$ minus the probability of the complementary event:

The number of ways to choose $4$ out of $24$ shoes is:


Choose the $1$st shoe out of $24$ shoes
Choose the $2$nd shoe out of $23$ shoes
Choose the $3$rd shoe out of $22$ shoes
Choose the $4$th shoe out of $21$ shoes


The number of ways to choose $4$ out of $24$ shoes with no pairs is:


Choose the $1$st shoe out of $24$ shoes
Choose the $2$nd shoe out of $22$ shoes
Choose the $3$rd shoe out of $20$ shoes
Choose the $4$th shoe out of $18$ shoes


So the probability of choosing $4$ out of $24$ shoes with at least one pair is:

$$1-\frac{24\cdot22\cdot20\cdot18}{24\cdot23\cdot22\cdot21}$$



Please note that I've essentially taken into account the order of the shoes.

If I chose not to take it into account, then I would need to divide each result by $4!$.

But since this factor appears in both the numerator and the denominator, I can ignore it.
To select a pair, the order is irrelevant.
1. Select one shoe
2. Select one shoe - chance this is a pair of first = 1/23 = .04
3. Select one shoe - chance this is a pair of first = 1/22 or chance this is pair of second = 1/21 = 1/22+1/21 = .09
4. Select one shoe - chance this is a pair of first = 1/21 or of second = 1/20, or of third = 1/19 = 1/21+1/20+1/19 = .15
After four selections, probability is .04+.09+.15=.28