What is the chance all choices are different? Suppose everyone (7 people) chooses independently and randomly out of 7 choices.
What are the odds that no two people have picked the same option?
I feel like the answer is $5040/823543,$ but I am rusty with anything math related.

The answer by whuber shows a nice way to approach this from first principles in combinatorics.  In this answer I will just note that this question is essentially asking for a probability from the classical occupancy distribution discussed in O'Neill (2020).  Given $n$ balls randomly allocated to $m$ bins, the probability mass function for the number of occupied bins $K_n$ is:
$$\mathbb{P}(K_n=k) = \text{Occ}(k|n,m) \equiv \frac{(m)_k \cdot S(n,k)}{m^n}.$$
In your case you have $n=7$ balls randomly allocated to $m=7$ bins, and your probability of interest is for full occupancy, which is:
$$\text{Occ}(7|7,7) = \frac{(7)_7 \cdot S(7,7)}{7^7} = \frac{7!}{7^7} = 0.006119899.$$
You can obtain values from the occupancy distribution in R using the probability functions in the occupancy package.  The present probability is easily obtained using the mass function for the occupancy distribution, using the query occupancy::docc(7, size = 7, space = 7).
Given your interest in determining the probability of full occupancy in this problem, you might be interested in the more general analysis of the occupancy number that is contained in the linked paper.  The occupancy distribution is quite interesting an it solves an antique problem in probability theory.  It also has a number of interesting generalisations.
