Answer: a.

Explanation:

A while is an unspecified period of time; awhile means for a while and is usually combined with an action. For example:

When I learned I was the victim of a data kidnapping exploit, I just sat and banged my head on the keyboard awhile.

Or I could say:

When I learned I was the victim of a data kidnapping exploit, it was a while before I stopped banging my head against the keyboard.

Here’s a good explanation of awhile vs. a while and some more examples of use from The Grumpy Grammarian:

The test of which to use is to consider whether “for a while” may be used in the sentence where we intend to place (or have placed) the word awhile – without changing anything else.

Examples:
> “I’ll wait here awhile” is correct because we could also say, “I’ll wait here for a while.”
> “I’ll wait here for awhile” is not correct because we have actually used the word for twice, given that awhile = for a while: “I’ll wait here for for a while.”
> “I’ll be there in awhile” is not correct because we would not say, “I’ll be there IN FOR a while.”
> “This may take awhile” is not correct because “This may take for a while” is not idiomatic English.
> “My mother is staying awhile” is correct because we could also say, “My mother is staying for a while.”

The two-word noun phrase (a while) is probably more often the correct choice than is the one-word adverb (awhile). Certainly, most misuses of a while / awhile involve using awhile where a while is the appropriate construction.

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Answer: b

Explanation: A while is a period of time; awhile means for a period of time, so “for awhile” would mean “for for a while.”

Taiichi Ohno studied supermarket operation for a period of time before developing Toyota’s rules for lean production.

Sherry Coven explains while vs. awhile on Everything Language and Grammar.

Maeve Maddox also covers the subject on Daily Writing Tips.

Follow us on Twitter @tao_of_grammar.