Learn English – “She noticed the change of (in?) her son” – choosing the right preposition

In general, you do not need a preposition in such cases. #3 is correct: "I have no idea what word to use."

I'm not sure what the general rule is or if there is one -- this is a pretty broad example. I think, though, that you don't need a preposition if what follows "I have no idea" (or similar phrases, of course) includes a verb, but you do need a preposition if it does not.

I have no idea where to go.

I have no idea how to fly an airplane.

But:

I have no idea ABOUT that.

I have no idea OF my own.

I have no idea BEYOND Sally's last suggestion.

There's a problem with this question, and these kind of tests. Any of the five choices make grammatical sentences in English.

One can argue that C is the best answer, as it talks about one action of running:

"run along the path, up the steps and into one's house."

But B is not incorrect, as one can

"run along the path, towards the steps and into one's house."

Maybe the steps are a landmark that shows the direction of the house.

Likewise, A, D and E do not make ungrammatical sentences. But one assumes they are not examples of the kind of sentence and use of prepositions that the test is testing.