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I am always confused between "a number of programs" and "several programs" for saying "many programs". Google Ngrams show that both are widely used terms. Could someone tell me which one should be used when?

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Do you mean that you're not sure when to specify a number (three programs) or write several programs ? –  aedia λ Sep 21 '11 at 22:54
    
@aedia-- You can say "I've written a number of programs". I think this may be confusing the OP. –  simchona Sep 21 '11 at 22:56
    
@aedia: No. "a number of programs" vs "several programs". –  samarasa Sep 21 '11 at 23:05

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They both mean that more than one thing is involved, but...

  • several indicates a small number: "more than a few; an indefinite small number". It falls somewhere between "a few" or "some", and "quite a few".

  • a number of tends to imply more uncertainty, or at least flexibility, about the exact number involved. In other words, you see it in places where the count really does not matter:

Problems may be subdivided in any of a number of ways...

...suggested a number of ingenious ways that...

But if you put quite in front of it... "quite a number of" means that the count really does matter, and you are asserting that there are a lot.

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Ah well. If you drop that quite into "I've written a few programs", I think you change it from an admission that you haven't written very many programs at all really, into a bold assertion that you've probably written far more than would be expected (always assuming you don't follow it with ", but...", of course). That's quite a word, that "quite". –  FumbleFingers Sep 22 '11 at 1:52

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