Probability of getting a pair of socks from a drawer if three are drawn I'm really struggling with this concept, hoping you guys could help me out.
Question: You have been provided with 20 pairs of socks within a box consisting of 4 red pairs, 4 yellow pairs, 4 green pairs, 4 blue pairs and 4 purple plairs.
The pairs have been separated out and you must take out a pair of socks.
Consider these problems and provide a calculation for each: 


*

*Probability of drawing a matching pair if you randomly draw 2 socks?

*Probability of drawing a matching pair if you randomly draw 3 socks?

*(Repeats up to randomly drawing 5 socks)


For 2 socks I got the following:
40 possible socks * 39 other possible socks = 1560 possible combinations of socks / 2 (to remove duplicate matches) = 780
For each set of socks, there are 8. 8 * 7 (7 other socks to each being matched) = 56 possible combinations in each set of socks / 2 to remove duplicates = 28 possible combinations of socks in each set.
28 / 780 = 0.036 probability of drawing a pair when drawing 2 socks from the drawer.
I'm completely lost when it comes to drawing three socks from the drawer, however - 
Cheers guys!
 A: To pull two matching socks in two draws.
$\frac{5{4\choose2}}{20\choose2} = \frac{5*4*3}{20*19} = \frac{3}{19}$
To pull two matching socks in three draws.
you can pull 3 socks of the same color, or 2 socks of one color, and sock of annother color.
$\frac{5{4\choose2}*16 + 5*{4\choose3} }{20\choose3}$
we can keep going with this methodolgy to work up to 4 and 5
4 draws.
$\frac{5{4\choose2}*16*12 + 5*{4\choose3}*16 + 5*{4\choose4} }{20\choose4}$
5 draws.
$\frac{5{4\choose2}*16*12*8 + 5*{4\choose3}*16*12 + 5*{4\choose4}*16 }{20\choose5}$
However at 5 draws, it is easier to thing of the probablility of not getting a match.
$1 * \frac{16}{19} *\frac{12}{18}*\frac{8}{17}*\frac{4}{16}$
A: HINT: with 3 socks you must find the probability that all socks have different color, the probability that 2 socks have the same color, and the probability that all socks have the same color. Name these probabilities as P1, P2 and P3... with the number referencing the different amount of different colors.
For any number of socks there is the same strategy: probability for different amount of different colors, i.e. for 5 socks you want to know P1, P2,..., P5.
After you do that then you must multiply each probability for the probability of get at least one sock of one of the colors for this group. "At least one" is the opposite to "no one" but this last probability is easier to calculate, after you get the complementary that is the probability that you are searching.
After you did that you sum all these multiplications and voilá!, its done.
A: I struggled with the above explanations so came up with the following: 40 socks total, 8 socks per colour.


*

*2 draws


I draw one red: chances are 8 / 40
I draw a second red: 7 /39 => second draw is dependant from first so i need to multiply chances of the two draw: 8/40 * 7/39 = 56 / 1560.
Same reasoning applies for the other colours so I can multiply by 5 my previous result: 5 x 56 /1560 = 280 / 1560 


*

*3 draws


Using the red socks, i have the following combinations when drawinf three times:


*

*1 red then 1 red then 1 non-red

*1 red then 1 non-red then 1 red

*1 non-red then 1 red then 1 red

*1 red then 1 red then 1 red


probability will be respectively:


*

*(8/40 * 7/39 * 32/38) +

*(8/40 * 32/39 * 7/38 ) +

*(32/40 * 8/39 * 7/38 ) +

*(8/40 * 7/39 * 6/38 ) 

*= 1,792 / 59,280 * 3 + 336 / 59280 = 7,504 / 59280


All colours have the same numbers so we can multiply by 5 so: 5 * 7,504 / 59,280
Same work can be done for 4 and 5 draws although factoring it is tempting and quite easy.
A: 
You have been provided with 20 pairs of socks within a box consisting of 4 red pairs, 4 yellow pairs, 4 green pairs, 4 blue pairs and 4 purple plairs.

So $40$ socks in total.  $8$ of each among $5$ colours.


*

*Probability of drawing a matching pair if you randomly draw 2 socks?

*Probability of drawing a matching pair if you randomly draw 3 socks?

*(Repeats up to randomly drawing 5 socks)


It is easier to evaluate the probability for the complement: that of drawing no matching colours.
So the probability for drawing a matching pair when drawing two socks is:
$$p_2= 1-\dfrac{\dbinom 52\dbinom 81^2}{\dbinom{40}2}$$
And likewise for the rest.
