through and by

I sometimes wonder which one I should use, through or by. Any difference between the two sentences below? Are they both OK?

We supported the department by promoting the sales of the products.
We supported the department through promotitng the sales of the products.

"by" is better in that sentence.
Before an "ing" word, "by" is usually more idiomatic, and a better choice.

by promoting products
by jumping over the fence
by inventing a new method
by putting it in the fridge

You would certainly be understood, however, if you chose "through" instead.

I personally say "through" in this context almost exclusively when followed by a noun phrase:

through the use of hand signals
through sales promotions
through no fault of his own
through the grapevine (idiom for "word of mouth")