In our garden live a couple of lizards which can change thei

In our garden live a couple of lizards, which can change their color. I made the following observations:

1. There are 5 (distinguishable) lizards: 2 live under the house, 2 in the old tree and 1 in the shed. They change their color only between gray, brown and green.

2. The 2 lizards under the house are always gray.

3. The 2 lizards in the old tree have an arbitrary color, but they never have the same color.

4. The lizard in the shed always has the same color as one of the other lizards.

How many combinations of colors can the 5 lizards in our garden have?

Solution

Lizards under the house can have colors in 3 ways as both have same color

Lizard in old tree can have color in :3*2 =6ways as first lizard can have 3 colors and second one must have one of remaining 2 colors

The lizard in shed can have the same color as one of the three colors: Either of lizards under house or two colors from lizards in old tree. So 3 colors to choose from. But then lizards only have three possible colors. So in case all lizards in old tree and lizards under the house have different colors then it makes no sense to choose a color for the lizard in shed since it can only have three colors so it will always have same color as one of the other lizards.

This case has: (3 ways to colors lizards under the house)*(2*1 to color lizards in old tree)=6 ways

Now consider the case where one of lizards in old tree has same color as the lizard under house.

3 ways to color lizard under house.

C(2,1)=2 ways to select one lizard from old tree to color same color as lizards under house.

2 ways to color other lizard in old tree.

So, 3*2*2=12 ways.

Now lizard in shed has 2 colors to choose from. This is done in 2 ways

So,12*2=24 ways

So total: 6+24=30 ways