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.

I agree with your solution and @whuber's elegant solution, here is my rational to arrive at the same solution by simply counting all options first, then taking the ratio at the end to compute the probability:

*

*Let's assume the set of possible choices is:
$ O = \{1,2,3,4,5,6,7\} $


*We can imagine 7 people picking numbers from this set $ O $ generates a sequence of these 7 numbers, for example 2547551 would be one of many options. All possible ways of arranging 7 numbers in a sequence where repetition is allowed is: $ 7^7 $. We allow repetition here because the problem states that people choose independently, so one person's choice is not going to affect other person's choice.


*The only way that no 2 person pick the same option would be all sequences where no repetition is allowed. One possible option would be 1234567. Number of possible arrangements of 7 numbers with no repetition is 7!


*So the probability that no 2 people picked the same number is:
$ \frac{7!}{7^7} $
You can also run a simulation and arrive at the same solution if you have doubts.  I used Python.

*

*Generated all possible permutations of these 7 numbers, counted them, assigned the result to a.

*Counted the options where no number in a sequence was repeated, assigned the result to b.

*Computed b/a

Here is the Python program and its output in case anyone interested:
import itertools

O = [1,2,3,4,5,6,7]
a = 0
b = 0
all_combinations = itertools.product(O, repeat=len(O))
for sequence in all_combinations:    
    a +=1
    if len(set(''.join(map(str,sequence)))) == len(O):
        b += 1  
    
print(f"Number of all possible ways of arranging 7 numbers with repetition = {a}")
print(f"Number of all possible ways of arranging 7 numbers without repetition = {b}")
print(f"Probability of having no 2 people selecting the same option = {b/a}")

>>> Number of all possible ways of arranging 7 numbers with repetition = 823543
>>> Number of all possible ways of arranging 7 numbers without repetition = 5040
>>> Probability of having no 2 people selecting the same option = 0.006119899021666143

