FOR

Use 'for' preceded by the following nouns:

a check for (amount of money)

Example: She gave me a check for $50.

a demand for something

Example: Unfortunately, there wasn't enough demand for our product.

a need for something

Example: There is a real need for discipline in this class.

a reason for something

Example: I have a reason for doing that!

IN

Use 'in' preceded by the following nouns: 

a rise in something

Example: There has been a rise in prices recently. 

an increase in something

Example: We have seen many increases in production levels.

a fall in something

Example: There has been a fall in prices recently.

a decrease in something

Example: We have seen many decreases in production levels.

OF

Use 'of' preceded by the following nouns:

a cause of something

Example: She is the cause of all his problems.

a photograph OR a picture of something or someone

Example: He took a photograph of the mountains.

TO

Use 'to' preceded by the following nouns:

damage to something

Example: I did a lot of damage to my car the other day.

an invitation to a celebration of some type

Example: We were invited to their wedding.

reaction to something

Example: Her reaction to his behavior was quite funny.

a solution to a problem

Example: He provided the solution to our financial situation.

an attitude to something (or TOWARDS something)

Example: Your attitude to your problems doesn't help them get resolved.

WITH

Use 'with' preceded by the following nouns:

a relationship with someone or something

Example: My relationship with Mary is wonderful.

a connection with someone or something

Example: His connections with the CIA are very limited.

a contact with someone or something

Example: Have you had any contact with Sarah?

BETWEEN

Use 'between' preceded by the following nouns:

a connection between TWO things

Example: There is no connection between the two crimes.

a relationship between TWO things

Example: The relationship between the two friends was very strong.

a contact between TWO things

Example: There is little contact between the two parents.

a difference between TWO things

Example: There is no difference between those two colors.