[Math] Combinatorics: How many ways to wear these socks.

combinatorics

Four socks in a drawer: two whites and two blacks. How many sock combinations can I wear?

It is obvious to me that the answer is four:

  1. Two white socks
  2. Two black socks
  3. White on the left foot, black on the right foot
  4. Black on the left foot, white on the right foot

However, I am having trouble devising the proof of this. If I start with the left foot, then I have four socks to choose from, but two are identical to others so I don't take them into account:

$$\frac4{1+1} = 2$$

Then I take the right foot, which has three to choose from but one is identical:

$$\frac31 = 3$$

Then $2+3=5$, which is not the right answer! What am I doing wrong here?

My goal is to know how to generalise. Next time I may have 20 different colours, and arbitrary numbers of each colour sock.

Thanks.

Best Answer

Your first answer is correct.

Trying to correct your second answer, you might have written $\dfrac{2}{2}+\dfrac{2}{2}=2$ distinct colour choices for the left foot and either $\dfrac{1}{1}+\dfrac{2}{2}=2$ or $\dfrac{2}{2}+\dfrac{1}{1}=2$ for the right foot (depending on which colour you had chosen for the left foot, though it makes little difference here).

Then, instead of adding, you should have multiplied $2 \times 2 =4$ to give the final result.

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