Shoe Dilemma

To solve this problem first we take the left shoe of each pair and distribute this among the $7$ people.This can be done in $7! = 5040$ ways.

Next we have to distribute the right pair such that none one gets a proper pair.So here we take the idea of derangements. The number of ways we can distribute the shoes so that no one gets a proper pair id $D_7 = 1854$ ,where $D_7$ is the derangenments of $7$ objects.

So the total number of possibilities is = $1854*5040 = \boxed{ 9344160}$ .

I solved this problem using this sum:

$\sum_{i=0}^{7} (-1)^i \binom{7}{i} (7 P i)((7-i)!)^2=9344160$

In each iteration, we will choose $i$ people that will get a proper pair of shoes. we have $\binom{7}{i}$ options to chose them, and $(7 P i)$ . options to chose shoes for them. For the other people, we need to choose one shoe for each leg, for each leg, we have $(7-i)!$ options to chose, i.e. $((7-i)!)^2$