Question: Distributing Coins

As I mention in the video, this question is out of scope for the GMAT. I wanted to use it to demonstrate the utility of listing and counting possible outcomes.

A "mathematical" approach would involve a technique known as "partitioning."

If you check the list, you'll see that {6,0,0}, {0,6,0} and {0,0,6} are already listed separately in the list of 28 outcomes.

To see why we can't use the FCP, let's see what happens when we start.

STAGE 1: Give Alex some coins
We can give Alex 0, 1, 2, 3, 4, 5 or 6 of the coins.
So, we can complete stage 1 in 7 ways

STAGE 2: Give Bea some coins
In how many ways can we complete stage 2?
It depends on how many of the 6 coins we gave to Alex.

- If we gave 0 coins to Alex, then there are 6 coins remaining, which means we can give Bea 0, 1, 2, 3, 4, 5, or 6 coins.

- If we gave 1 coin to Alex, then there are 5 coins remaining, which means we can give Bea 0, 1, 2, 3, 4, or 5 coins.

- If we gave 2 coins to Alex, then there are 4 coins remaining, which means we can give Bea 0, 1, 2, 3 or 4 coins.

- If we gave 3 coins to Alex, then there are 3 coins remaining, which means we can give Bea 0, 1, 2 or 3 coins.

Etc...

So, there's no way to determine the number of ways to complete stage 2. The same goes for stage 3.

So, we need a different approach.

To my knowledge, there's no nice way to solve this question using the Fundamental Counting Principle.

There is a technique, called the Separator Method, that we can use, but it's beyond the scope of the GMAT.

Cheers,
Brent

That's correct. However, the GMAT does not require us to know this rule.

For this question, we are simply listing and counting all of the possible outcomes.
So there's no need to introduce any other operations.

You are correct about the FCP. Since we are not using the FCP to answer the question, there's no need to multiply.