Possible outcomes of 6 components (each with 3 options)

[AnnaLouii]

Hi,

I'm trying to remember back to my a-level days and iterations but can't quite remember how to do it when each has 3 options.

My problem is:

How many possible outcomes are there for changes in 6 zone sizes? Each zone can increase, decrease or not change.

For example:

Zone 1 could increase but zone 2-6 remain the same size

OR

Zone 1,2,3 could decrease, zone 4 increase and zone 5 and 6 remain the same size.

Any ideas?

 

[pka]

My problem is:
How many possible outcomes are there for changes in 6 zone sizes? Each zone can increase, decrease or not change.

If you flip a coin six times then there are \(\displaystyle 2^6\) possible outcomes.
So what is this answer?

 

[JeffM]

Hi,

I'm trying to remember back to my a-level days and iterations but can't quite remember how to do it when each has 3 options.

My problem is:

How many possible outcomes are there for changes in 6 zone sizes? Each zone can increase, decrease or not change.

For example:

Zone 1 could increase but zone 2-6 remain the same size

OR

Zone 1,2,3 could decrease, zone 4 increase and zone 5 and 6 remain the same size.

Any ideas?

This is why math is not memory. We can and do forget formulas, but, we hope, not logic.

I have one object that can be in just one of k possible states. So I have k distinct possibilities.

I have two distinct objects, each of which can be just one of k distinct possible states. For any one state that the first object happens to be in, how many possibile states can the second object be in? Yes, k. And the first object has k possible states. So the total number of possibilities is k * k.

I have three distinct objects, each of which can be in just one of k distinct possible states. For any one pair of states that the first two objects are in, how many possible states can the third object be in? Yes, k. But we previously figured out that the number of possible pairs is k * k. So our answer is k * k * k.

So if we have three distinct zones (objects), each of which is in one of three distinct states (growing, decreasing, or stable), how many possibilities are there?

If you want to check, you can list them

1G, 2G, 3G
1G, 2G, 3S
1G, 2G, 3D

1G, 2S, 3G
1G, 2S, 3S
1G, 2S, 3D

I shall let you finish up.