Question: Distributing Coins

Comment on Distributing Coins

Can we solve this sum through some different methodology, given the method illustrated will not be feasible in exam?
gmat-admin's picture

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."

I thought that we need to consider 28 times 3 because the option (6,0,0) is different from (0,6,0) and from (0,0,6)... Where is my mistake? tks
gmat-admin's picture

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.

Hello, Didn't understand why can't we use fundamental counting principle here? This is three stage activity and for each activity we need to distribute 0 to 6 coins. confused :(
gmat-admin's picture

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.


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.

Thanks, understood.

Thank you! I had the same doubt.

Hi Brent
Can we solve this question using fundamental counting principle?

gmat-admin's picture

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.


in probability theory from my university I learned you can use n+r-1 choose r-1 to solve this problem.
gmat-admin's picture

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

Hi Brent, not sure why is not multiplication of each outcome but addition for total combinations here? Is this because it's not based on FCP? Getting a bit confsued. Thanks
gmat-admin's picture

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.

Get it, thanks Brent


Since the question mentioned that there are "6 identical coins be distributed". Can I understand identical as that these coins look different, so the question will be even more complicated.
gmat-admin's picture

I'm not sure what you're asking.
If all 6 coins are identical, then, for example, there's exactly 1 way Alex, Bea and Chad can receive 2 coins each.
If we say the 6 coins are all different, then there would be 90 ways Alex, Bea and Chad can receive 2 coins each.

So, yes, if the 6 coins are different, question becomes more complex, but that's not what the question is asking.

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