At least some of these locks are order-independent, which reduces the solution space even further. That is to say, 1 2 3 is the same combination as 1 3 2. Though I think that this sort usually has ten buttons, rather than 5.
You’re thinking of the Kidde Key Safe, formerly sold under the brand names GE and Supra. Those lock boxes have ten buttons instead of five and order does not matter. 1 2 3 is the same as 1 3 2. The total number of combinations on a Kidde is 1023 if you don’t count the null combination of pushing no buttons at all. For each of the ten buttons, you can either choose to include it in the combination or not include it, so the number of choices is 2 to the 10th power, 1024, but that includes the null combination. The instructions recommend that you use at least four buttons in your combination and most users choose a combination with exactly four buttons. In that case, the number of possible combinations is 10x9x8x7 dived by 4x3x2x1 which is 210.
The locks described upthread are called Simplex mechanical locks, sold under the names Kaba/Ilco/Unican. Order does indeed matter on a Simplex lock; 1 2 3 is not the same as 1 3 2.
There is an electronic version of the Simplex lock which allows the same button to be used more than once. For example, 1 3 1 3 1 is a valid combination on the electronic version but not on the mechanical version. A mechanical Simplex can be upgraded to an electronic one by replacing the “combination chamber”.
I once visited a frat house with a Simplex lock on their front door. The combination was the street address of the building. I guess they wanted something so simple they could remember it even when drunk.
Mine were a little different from Meacham’s. The numbers went up to at least 44, maybe 48. I had two such locks, one for my wardrobe locker and one for my gym locker. One of them used 7-43-36 and the other 9-41-36. I quickly learned that 8-42-36 would open them both. I then forgot which was which.
It also helped that the combo was written in 2" high contrasting digits somewhere near the door. Number skills are one of the first to go when drunk. Remembering the route to crawl back to the cave and recognizing numerals are two lower-level skills that last further towards unconsciousness.
Don’t ask me how I know this; I no longer remember. 
I had the master list of combinations at my high school - I was the trusted computer kid that knew how to make the office staff’s life easier, so I was the one doing printouts for everyone. Although I don’t remember the details now, I do remember a definite, specific pattern in the combinations.
Considering the type of buildings that tend to use them - I’ve mostly seen them in dorms - you wouldn’t have time to try very many guesses before someone noticed you.
At my high school, the locker combinations came in lots of 40. If one student’s combo was 9-32-12, then another’s was 10-33-13, and another was 11-34-14, and so on all the way around the dial. I suspect that they set the internals in the factory, and then just rotated the dial before attaching it.
Meanwhile, of course, it was also easy to find the last digit of any given person’s combo, because it’d shift by the same amount every time it was opened, and most didn’t bother to spin the dial after that. So if you knew two or three of the patterns, and saw someone’s lock after they had used it, you had a pretty good chance of figuring out anyone’s combo.
… which means you’d either have to choose a time of day when nobody’s there, or you could just walk up and try two or three combinations then pretend to answer a cell phone call and walk away planning to come back tomorrow to try a few more, or you could just hang back pretending to read your email and watch several people go through the door and see what buttons they pushed.
Even supposing you can’t see the keypad while they push the buttons, just pay attention to what you can see. Suppose you can clearly see that everyone is pushing three buttons in sequence (do simultaneous button pushing) and the first button seems to be the very top one and after that you can’t see clearly but it looks like the third button is higher up than the second one. Now you know the combination is either 1-3-2, 1-4-2, 1-4-3, 1-5-2, 1-5-3, or 1-5-4. Walk up and try the first three. Then pretend to answer your cell phone and walk away for five minutes. Then go back and try the next three.
Despite these challenges, this type of lock is still much less secure than an ordinary keyed lock using pin tumblers.
Now, if this door merely gets you into a hallway where all the interior doors are locked, with commercial grade 6-pin locks, then that doesn’t accomplish much. Likewise, if the only thing behind the door is a storage closet with buckets and shovels then who cares. What I said was don’t put it on the front door of your house.