Skin color in a fish is inherited via a single gene with four different alleles. How many different genotypes would be possible in this system?

1. Break Down the Problem

The problem involves determining the number of possible genotypes for a situation where a single gene has four different alleles. To find the total number of genotypes, we need to consider how alleles can combine in pairs in a diploid organism.

2. Relevant Concepts

A diploid organism has two alleles for each gene, and we will utilize the concept of combinations to find all the possible pairs of alleles. The total number of different alleles is given as four (let's denote them as A, B, C, and D).

3. Analysis and Detail

Since there can be two types of combinations for alleles:

1. Homozygous Genotypes: These occur when both alleles are the same.

   • Possible homozygous genotypes: $AA, BB, CC, DD$ (4 genotypes)

2. Heterozygous Genotypes: These occur when the two alleles are different.

   • Here, we need to calculate the number of combinations of the available alleles taken 2 at a time. The formula for combinations is: $C(n, k) = \frac{n!}{k!(n-k)!}$ where $n$ is the total number of alleles and $k$ is the number of alleles chosen at a time. For our case, this is:
   • $C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6$

   The possible heterozygous genotypes are: $AB, AC, AD, BC, BD, CD$.

4. Verify and Summarize

Now, we sum the number of homozygous and heterozygous genotypes:

• Homozygous: 4
• Heterozygous: 6

Thus, the total number of different genotypes is: $4 + 6 = 10$

Final Answer

The total number of different genotypes possible with four different alleles is 10.