If you have 8 socks and 3 pairs of shoes how many sock shoe combinations can you wear?

Answer 1

For 8 socks...

Each shoe pair needs two socks. For the first foot, you can choose from 8 socks. For the second, you can choose from 7. Therefore, for each pair of shoes, you have 7*8 possible sock combinations, or 56 possibilities per pair of shoes. With three pairs of shoes, this means it's 3*56, or 168 possible combinations, assuming that the shoes can only go together in pairs.

If the shoes can be unpaired, but must be on the correct feet, then there is a slightly more complicated answer. You have three shoe options per foot. for each shoe you choose, you have another three options for the other foot. Making just your shoe combinations 3*3, or 9. For the first shoe you wear, you can choose from 8 socks. So for the first foot, you have 3 options for shoe, and 8 options for socks, leaving you with 27 combinations for that foot. For the second foot, you have 3 options for shoes, since they don't have to match. for each of those three options for shoe, you have 7 options for socks. Therefore, you have 21 potions for the second foot. For each first foot option, any of the second foot options can be selected, making your answer 27*21 or 567 combinations.

If you can wear the shoes on the wrong feet...

This means that foot #1 has 6 options for shoes, and foot #2 has 5 (since you can't wear one shoe on both feet).

For the first shoe, you have 8 socks to choose from, so your options are 6*8 or 48 possible combos on the first foot. For the second foot, you have 5 options for shoes, and each of those has 7 options for socks. Thus you have 5*7 or 35 possible combinations of sock and shoe for second foot. For each combo on foot 1 we have 35 combos on foot 2 to choose from, which means 48*35 is our answer, or 1,680 possible combinations if we don't have to wear the shoes on the correct feet.

If you're in college or beyond, I would go with either 567 (meaning you can't wear the shoes on the wrong feet, but they don't have to match) or 1,680 (meaning you can wear them on the wrong feet). If you're in high school or below, go with 168 because they probably didn't try to mix you up with mixed shoe pairings. At least, I don't think they would. But you never know.