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Much is used with uncountable nouns in negative sentences and questions:
• How much milk is there in the fridge?
– Not much.
Many is used with countable nouns in negative sentences and questions:
• How many eggs are there?
– There aren’t many.
Many can be used in positive sentences. However, this sounds more formal:
• The prime minister has many supporters.
A lot of or lots of (more informal) can be used in positive and negative sentences and questions with countable and uncountable nouns:
• We don’t eat a lot of potatoes but we eat lots of carrots.
Although these phrases are normally interchangeable, you can only use quite + a lot of (not:
quite + lots of):
• I have quite a lot of old CDs.