# Sorting 10 pairs of socks into 4 drawers

In how many ways can we sort 10 pairs of socks into four drawers without any additional conditions?

When I first saw this task I thought it was extremely easy, the answer must have been 10 x 9 x 8 x 7, however, this calculation does not take into account other possiblities, just like all pairs in one drawer.
Could you explain to me the easiest and the most intuitive way to solve this problem?

• Are the socks distinguishable? Are the drawers? If we can tell everything apart, then there are $4$ places each pair can go and these choices can all be made independently, so $4^{10}$. But perhaps you mean something else. – lulu Mar 23 '17 at 14:40
• Can you split the pairs into two indistinguishable socks? – Henry Mar 23 '17 at 14:43
• If the pairs of socks are not distinguishable, then your question amounts to finding the number of solutions of the equation $x_1 + x_2 + x_3 + x_4 = 10$ in the nonnegative integers, where $x_k$ is the number of pairs placed in the $k$th drawer. This is a combination with repetition. – N. F. Taussig Mar 23 '17 at 14:45
• What I have written in the first line is the whole question, I think the rest is up to my assumptions. – ILoveChess Mar 23 '17 at 14:45

Take the first pair of socks and decide where to put it, that gives you 4 options. Then proceed with the second pair. Independent of what you did with your first pair, you have 4 options. Then you take the third pair, and again you'll have 4 options since you don't need to care where you had put the first 2 pairs. Same for all the following pairs, which gives you $4^{10}$ possibilities (4 for each of the 10 pairs of socks).
Even if you put all $10$ pairs of socks into one drawer(being that there's four drawers) that's still four possibilities. Going thru the possible placement of each position of $10$ pairs of socks would just come out to $4^{10}$ possibilities.