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Which term should be used in the following sentence:

"with" ; or; "in" ? Or either one?

And why/why not? Please provide reasoning!

Thank you!

 " I do a course (in/with) the open university." 

First, let's address two issues with your sentence. Clarify two things:

" I do a course (in/with) the open university."

"the open university" - - 1st. Needs clarification.

"I do a course" - - 2nd. Needs clarification.

1st clarification: Stating "the open university" not capitalized is incorrect. Since this is a college, a college is a proper noun (LBH P.322) Capitalize the names of colleges (ie., The Open University.)

There's also two colleges with "Open University" as the Institution's name. Which is it?

Is it The Open University based in the United Kingdom, or is it the Open University in New Delhi?

If it's the Open University in New Delhi, any determiners, "a" "an" "the" that precede the name is not capitalized. (LBH P.254). Just say, the Open University.

If "the" is in the title of the University, then "the" is capitalized because it's part of the name of the college.(Ref: LBH P.254). Just say, The Open University.

There are 10s of 1000s of colleges. With this, there are colleges with like sounding names and more common than one would think. If you attend a college with a name easily confused with another, end all confusion by stating the physical location of the college when referring to it in writing or in speech.

Example: the Open University in New Delhi.

If it's the internet based The Open University, there is only one The Open University. No need to state the location; you can if you want to.

Just remember to capitalize proper nouns (names of people, months of the year, cities, countries, proper name of colleges, etc.)

2nd clarification:

"I do a course"

I (subject) do (verb) a course (object).

The subject is ok. The object is ok. The challenge is with the tense of the verb in the sentence you provided.

"Do" is an active verb. Change the verb "do" to "Present Continuous" = doing. Saying "doing" lets the listener know you are doing something right now.

A good website that talks to all verb tenses is at this site: https://www.grammarly.com/blog/verb-tenses/

With the grammar now mostly fixed, your sentence now looks like this:

I am doing a course (in/with) The Open University.

in vs. with - using in is incorrect, but it requires explanation as to why.

Yes, in is a preposition. "at" "in" or "on" but...rule states:

• 
  

  Use "at" before a specific place or address. (www.grammarly.com / Purdue University OWL. https://owl.purdue.edu/owl/purdue_owl.html/Little Brown Handbook P. 519) This rules out using in for your sentence.

  
  
  

  Your sentence now looks like this:

  
  

I am doing a course (at/with) The Open University.

Using "at" will work, but only for colleges that are at a fixed address or location (LBH P. 519/www.grammarly.com/Merriam-Webster/prepositions)

Some will still use at, so this is argumentative, and I'll leave the choice to you; but, since The Open University is internet based, and students don't actually physically go to a physical location at the Internet based college's headquarters to do their course work, but do it on the internet from anywhere at a computer, the college is not really at any one physical place, so using "at" is ruled out. Internet based colleges are somewhere out on the internet that's everyplace.

Still, using at can get you by, as this internet dynamic is now, and arguing this really doesn't help answer your question; but, it's not really accurate for Internet based colleges to use at. The only exception is if you're physically going to the college that sponsors an internet course and doing the course work from there. If you're doing a course at the Open University in New Delhi as well, and doing the course on campus, then use "at."

That leaves with. So, let's look at "with."

With is also a preposition; but, unlike at, with can be used for other relationships. It's categorized differently specifically for this kind of condition. (Little Brown Handbook, 11th ed. Pg.243 Pearson Education, Inc. 2010 Book.)

If we used "with," your sentence looks like this:

I am doing a course with The Open University. //Correct//

Using either "at" or "with" are both correct for colleges; but, for a college on the internet that is at no physical location, use the "other relationships" preposition of "with" not "at."

Hope this helps.