Learn English – Third alternative to “between” and “among?”

GrammarGirl did a whole post on this. She explains that there is a difference between between and among. She writes:

Here's the deal: you can use the word “between” when you are talking about distinct, individual items even if there are more than two of them. For example, you could say, "She chose between Harvard, Brown, and Yale" because the colleges are individual items.

She goes on to explain that in the following cases, you use the two slightly differently:

Relationships: The Chicago Manual of Style describes these as one-to-one relationships. Sometimes they are between two items, groups, or people, as in these sentences:

• Choose between Squiggly and Aardvark.
• Let's keep this between you and me.

Other times they can be between more than two items, groups, or people as in these sentences:

• The negotiations between the cheerleaders, the dance squad, and the flag team were going well despite the confetti incident.
• The differences between English, Chinese, and Arabic are significant.

On the other hand, you use “among” when you are talking about things that aren't distinct items or individuals; for example, if you were talking about colleges collectively you could say, "She chose among the Ivy League schools." If you are talking about a group of people, you also use “among”:

• Fear spread among the hostages.

• The scandal caused a division among the fans.

• Squiggly and Aardvark are among the residents featured in the newsletter.

Part of a Group: “Among” can also indicate that someone is part of a group or left out of a group, as in these examples:

• He was glad to find a friend among enemies.
• She felt like a stranger among friends.
• Sylvia was later found living among the natives.

From this, you were correct in your use of between. "Cook et. al." formed part of a distinct group of individuals, so you would use between.

The correct answer: "The successful collaboration between the contractors, the LA municipal office, and the funding provider resulted in reduced cost and higher efficiency."

Using between or among in conjunction with collaboration doesn't change the usage.

Your sentence draws a direct relationship between the specific parties, so use between.

To clarify further, you would use among if the sentence read, "The collaboration among invested parties resulted in reduced cost and higher efficiency."

You might find it helpful in the future to visualize your sentences as sketches.

The sentence with between: a picture with lines drawn between the parties. The relationship is defined.

The sentence with among: a dotted circle around the parties, lumping them together. Collaboration exists somewhere in that dotted circle but it's more ambiguous.