If a given population of diploid organisms contains only three alleles of a particular gene (say, W1, W2, and W3), how many different diploid genotypes are possible in the population?

To determine the number of different diploid genotypes possible from three alleles (W1, W2, and W3) of a particular gene, we can use simple combinatorial mathematics. In a diploid organism, each individual has two alleles for any given gene—one inherited from each parent.

Since there are three different alleles, and each organism has two alleles, we can calculate the potential combinations where the order does not matter (since W1W2 is the same genotype as W2W1). This is a problem of combinations without repetition, which can be calculated using the binomial coefficient, often represented as "n choose k," where n is the total number of items to choose from, and k is the number of items to choose.

The formula for the number of combinations is:

number of combinations = (n!)/(k!(n - k)!)

In our case, n = 3 (the number of alleles) and k = 2 (since we are looking for combinations of 2 alleles to create genotypes). Substituting the numbers, we have:

number of combinations = (3!)/(2!(3 - 2)!) = (3 × 2 × 1)/(2 × 1 × 1) = 3

However, this only accounts for the combinations of two different alleles. We also need to include the genotypes that have two copies of the same allele, which are the homozygous genotypes: W1W1, W2W2, and W3W3.

Adding these to our combinations, we have 3 from the heterozygous combinations (W1W2, W1W3, W2W3) plus 3 for the homozygous ones (W1W1, W2W2, W3W3), giving us a total of 6 different possible diploid genotypes.