There are 5 pairs of socks in a bag, so a total of 10 socks. Each pair of socks is of a different color from the others. After having randomly drawn 4 socks from the bag, what's the probability of picking 4 socks of different colors?
My answer was to first get the total number of possible draws as ${10 \choose 4} = 210$.
Then I'd proceed to compute the number of possible pickings of 2 pairs of socks out of the 4 draws:
${4 \choose 2} {4 \choose 2} = 36$.
Similarly, the number of pickings of 1 pair of socks + 2 loose socks would be ${4 \choose 2} {4 \choose 1} {4 \choose 1} = 96$.
The probability of picking 4 socks of different colours would then be the complementary set of events to the sum of the previous 2: $P($ no matching pair $) = 1 – \frac{36 + 96}{210} = 0.372$.
However the answer to the problem points to a different result. I understand the fact there are 5 colors should factor in somewhere, I only considered the number 4 (the draws) in my combinatorics and I guess this is the source of my mistake.
Best Answer
I think you do not need to do these cumbersome calculations .Let me show you another approach. Lets show the socks using boxes , each adjacent boxes represent same color.
$$\color{red}{\fbox{1}\fbox{2}}\quad\color{blue}{\fbox{3}\fbox{4}}\quad\color{green}{\fbox{5}\fbox{6}}\quad\color{orange}{\fbox{7}\fbox{8}}\quad\color{purple}{\fbox{9}\fbox{10}}$$
Now assume that we have some stickers which we use to select which sock will be selected. When we put these stickers onto these boxes , it will mean that we will select that box ,i.e, that sock.
In the beginning , we can select any of them , so we have $$\frac{10}{10}=1$$
Assume that we select the box $2$ , then we cannot select box $1$ , because all selected boxes must be different colors. Then , we have $8$ choices among $9$ possible boxes such that $$\frac{8}{9}$$
Now ,assume that we put the sticker onto the box $5$ ,so there left $6$ possible choices among $8$ boxes , so $$\frac{6}{8}$$
Assume that assume that the sticker has been put onto box $10$ , now we have $4$ suitable choices among $7$ , so $$\frac{4}{7}$$
Now , it is the time for finding their product such that $$1 \times \frac{8}{9}\times \frac{6}{8}\times \frac{4}{7 }= \frac{24}{63}=0.380952381$$