Problem is a countable noun and describes something that causes trouble or difficulty. We talk about having a problem or having problems with something, not about having a trouble. Compare the following:

• I've got a big problem with my computer. Can you come and have a look at it?
  (NOT: I've got a big trouble with my computer. Can you come and have a look at it.)
  
  
• I can't meet him in Paris and he can't meet me in London. It's a real problem.
  (NOT: I can't meet him in Paris and he can't meet me in London. It's a real trouble.)

We also talk about mathematical problems and solving problems of various sorts. Trouble cannot be used in this way:

• Children with learning difficulties find mathematical problems impossible.
  
  
• We couldn't solve the problem of getting across London in less than two hours.

With the verb cause, we can use both trouble and problems, problem as a countable noun and trouble as an uncountable noun. Compare the following:

• The recent football hooliganism in Sunderland caused the police a lot of trouble.
  
  
• The current drought is causing serious problems for the farmers in this area.

No problem! - What's your problem?

We also have the expressions No problem! which we use to say that we will be happy to do something or are happy for something to happen and What's your problem? which we use in a threatening way to ask someone about something we disapprove of. Compare the following:

• Could you look after Jimmy for me for five minutes while I pop out to the shops? ~ No problem!
  
  
• I'll finish this off tomorrow, if you don't mind. ~ No problem.
  
  
• I don't like people wearing face jewellery. ~ What's your problem? It's quite harmless.
  I think it could cause health problems in later life. ~ Well, that's their problem!
  

trouble

Trouble is mainly used as an uncountable noun and describes problems, worries or difficulties. Trouble can also be used as a verb. Compare the following:

• I'm having trouble with the printer now. Can you come and have a look at it?
  
  
• I'm a bit deaf and I had trouble hearing what she said as she spoke very softly.
  
  
• Why are you crying? What's troubling you? ~ It troubles me that I haven't heard from him for five weeks.
  
  
• I'm sorry to trouble you, but could you move your car forward a bit. It's blocking my drive.

In addition to cause, the verbs that the noun trouble collocate with include the following: put to, take, go to, save, get into, run into, and be in. These verbs cannot be used with problem in the same way. Compare the following:

• I'm sorry to put you to all this trouble ~ It's no trouble at all!
  
• I'm going to take the trouble to bake my own bread, rather than buy it from the shop.
  
• If you buy a dishwasher, it will save you the trouble of washing your dishes by hand.
  
• We ran into trouble as soon as we reached the motorway. It was jammed all the way from Epping to Cambridge.
  
• I shall get into real / big trouble, if I lend you my brother's bike.
  
• I was in serious trouble. I had run out of water and was still ten miles from the nearest oasis.

No trouble!

Note that the expression No trouble! is used in a similar way to No problem!

• I'm sorry to have kept you waiting for so long ~ That's no trouble!

problem / trouble + adjs

Note from the examples above that the adjectives big, real and serious collocate with both trouble and problems. Note that fundamental, insoluble and intractable collocate only with problem:

• A fundamental problem in the design of this car is the transverse engine.
  
• It was an intractable / insoluble problem. There was no way out of it.
  

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