Learn Hardcore French: Sans la souris, je ne peux pas terminer le programme. - Without the mouse, I cannot finish the program.

You cannot form du (that’s de + le) or saun (non-existent).
An article is only dropped after sans when you speak of something completely indefinite or uncountable: sans argent (without money), sans eau (without water).
Here, la souris is a specific, countable item, so you keep the definite article.

Why is la souris feminine and singular, and how can I tell it refers to just one mouse?

In French, souris (both the animal and the computer mouse) is a feminine noun: une souris, la souris. The article la signals singular feminine. If you meant more than one, you’d say les souris. Note that the written form souris is identical for singular and plural, but the article (la/les) and context tell you which it is.

Why do we use je ne peux pas instead of just je peux pas or je ne peux?

Standard French expresses negation with two parts around the conjugated verb: ne before and pas after. So je ne peux pas literally = I am not able.

In casual speech, people often drop the ne and say je peux pas, but in writing and formal contexts you must keep both.
Omitting pas (i.e. just je ne peux) sounds incomplete or overly formal.

Why is terminer in the infinitive and not conjugated?

After a modal verb like pouvoir (to be able), the following verb remains in the infinitive. You conjugate pouvoir (je peux) and leave terminer as terminer. This is the same pattern as je veux manger, tu dois partir, etc.

Why is there a definite article le before programme? Could we say terminer programme?

French nouns generally require an article (definite, indefinite, or partitive). Here you refer to a specific program, so you need le programme.

terminer programme is ungrammatical.
If you meant any program in general, you’d use an indefinite article: terminer un programme = to finish a program.
To finish this particular program, it must be le programme.

Could I use finir instead of terminer here? What’s the difference?

Both mean to finish, but they behave differently:

terminer directly takes an object: terminer un programme.
finir often requires de + infinitive when followed by another verb: finir de coder. You can still say finir le programme, but if you switch to an infinitive you’d say finir de programmer. In this context, terminer le programme is more straightforward.

Why is it la souris and not ma souris if it’s my own mouse?

French often uses the definite article when possession is obvious from context. la souris naturally means the (my) mouse here. If you want to stress that it’s your own, you can say ma souris, but it’s not required when the reference is clear.

How do I pronounce the final letters in peux, pas, and souris? Are they silent?

Yes, the final letters are silent:

peux (spelled -eux) → pronounced [pø]
pas (ends in -as) → pronounced [pa]
souris (ends in -is) → pronounced [suʁi]
French orthography often shows letters that aren’t pronounced; you rely on the article or verb form to know the number/gender.

Can I move sans la souris to the end of the sentence? Does it change the emphasis?

Absolutely. Both are correct:

Sans la souris, je ne peux pas terminer le programme. (emphasizes the lack of mouse)
Je ne peux pas terminer le programme sans la souris. (emphasizes the action of finishing)
The core meaning stays the same; you only tweak what you put in focus.

How would I say the same thing if the subject were nous or il?

Just change the form of pouvoir:

Sans la souris, nous ne pouvons pas terminer le programme.
Sans la souris, il ne peut pas terminer le programme.
Everything else (word order, articles, infinitive) remains identical.