Rock Lizard Colour Combinations: Calculation

The problem requires finding the number of ways to choose 2 different species (and thus colours) from a group of 9 distinct species. Since the order in which the lizards are chosen does not matter, this is a combination problem.

Identifying Combination Formula

The formula for combinations is used to find the number of ways to choose a subset of items from a larger set without regard to the order of selection. It is denoted as $C(n, k)$ or $\binom{n}{k}$ and calculated as:

$ C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} $

Where:

• $n$ is the total number of items to choose from.
• $k$ is the number of items to choose.

Applying the Formula

In this question:

• The total number of different lizard species (colours) is $n = 9$.
• The number of lizard species sitting on the rock at any time is $k = 2$.

Substitute these values into the combination formula:

$ C(9, 2) = \binom{9}{2} = \frac{9!}{2!(9-2)!} $

$ C(9, 2) = \frac{9!}{2!7!} $

Calculate the factorials:

$ C(9, 2) = \frac{9 \times 8 \times 7!}{ (2 \times 1) \times 7!} $

Cancel out $7!$:

$ C(9, 2) = \frac{9 \times 8}{2 \times 1} $

$ C(9, 2) = \frac{72}{2} $

$ C(9, 2) = 36 $

Final Answer

The number of possible colour combinations of rock lizards seen together on a rock is 36.